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Group theory and point groups
3 Group theory and point groups Yet it is natural for man, more than for any other animal, to be a social and political animal, to live in a group. St Thomas Aquinas (1225-1274): On Kingship
3.1 INTRODUCTION: GROUPS AND GROUP THEORY In this chapter we shall discuss the topics of group theory and point groups. We may look upon group theory as a tool for handling symmetry operations and their combinations in a quantitative manner, leading to an elegant procedure for describing the role of symmetry in chemistry. A point group provides a concise description of the symmetry of a finite body, and all finite bodies may be classified in terms of their point-group symmetry. The set of symmetry operations comprising a point group constitutes a mathematical group. 3.2 WHAT IS GROUP THEORY Group theory deals with sets of operations having the property that, when two operations of the set are carried out successively, the result is that which would be obtained by another single operation of the set, starting from one and the same initial situation. A set of operations forms a group provided that it satisfies this condition together with a number of group postulates. 3.2.1 Group postulates A group consists of a set of mathematical objects that may be symbolized by E, A, B, C,..., called members (also known as 'elements') of the group. In our discussion of group theory the members of the group will be, normally, symmetry operations, and they will be written in bold type. Thus, a symmetry group may be written collectively as 6{E, A, B, C, ...}, and all groups are governed by the following group postulates. Closure The combination of any two members of a set is also a member of the set; this group property is referred to as closure, and is implicit in Figure 1.4. The combination of the members is termed their product irrespective of the nature of the combination process; it will generally, but not necessarily, be multiplication. We may write a product as BA=C, (3.1)
Sec. 3.2]
What is Group Theory
39
where C is a member of the set A, B, ..., and we need to indicate the law of combination. If the law of combination is multiplication, then the group is said to be closed with respect to multiplication. Laws of combination In the case that the law of combination is multiplication, we have shown in Section 2.5.3 that the order of multiplication, normally, is important: if (3.1) represents a combination by multiplication then, in general, the reverse order of multiplication leads to a different result: AB=D,
(3.2)
but where D is also a member of the set A, B, C, .... Other laws of combination are inter alia addition, matrix multiplication and vector addition. Association The associative law (see also Section 2.2) holds for multiplication, that is, A(BC) = (AB)C.
(3.3)
IfBC = Z, and with (3.2), it follows that AZ= DC.
(3.4)
Identity member The group contains the identity, or unit, member E (see also Section 2.2.7), such that (3.5) AE = EA = A for each member of the set. Inverse member Each member of the set has an inverse (see also Section 2.5.3) that is also a member of the set; thus, (3.6) We see that these postulates are paralleled by the operations of matrices: this property will be taken up in later chapters, where we can introduce notation to distinguish between a symmetry operation and the matrix used to represent its action. 3.2.2 General group definitions a) A group may be finite, the number of members of the group being the order h of the group, or infinite. Thus, 6{E, A}, A 2 } is finite, but 6{E, AI, A 2 , ... , oo} is infinite; infinite groups have an important role in space-group theory. b) An abstract group is concerned only with the relationships in the set of operations, there being no specific interpretation attached to the members of the group. Thus, 6{E, A}, A 2 } could be a finite, abstract group, since no meaning is attached to the members other than E. c) A group in which all the members commute (see Section 2.5.3) is called an Abelian, or commutative, group. Thus, the group 6{E, AI, A 2 } , where EA i =
40
Group Theory and Point Groups
[Ch.3
A;E (=A;) andA;Aj=A;1; (i,j = 1, 2), is an example ofa commutative group. It is possible for a group to be nonAbelian although some of its members commute; point group D4., which we shall meet later, is one example. d) A group consisting ofa single member A and its powers zl' , A 3 , •.. , AP, is called a cyclic group of order p, where p is the smallest integer for which AP = E. Figure 2.1 illustrated the cyclic group C4 of order 4, containing the symmetry operators
C 4, C ~ (= C 2 ) , C ~ (= C 41 ), C: (= E). We see that operating with C 4 twice in
succession takes us from (a) to (c) via (b), and that operating with C 2 on the initial state takes us from (a) to (c) directly; thus, C2 is properly a member of the group C4 . We note also that the symmetry element C4 is associated with more than one symmetry operation, as discussed already in Section 2.2. e) A subset of members of a given group forming another group is a subgroup of the given group; the given group may, in turn, be termed a supergroup of the subgroup. Thus, a group consisting of the members C 2 (= C 21 ) and C ~ (E), is a cyclic group of order 2, C2 , and a subgroup of the group C4 which, itself, is a supergroup of C2 . f) Every geometrical body, crystal or molecule in our applications, may be characterized by a group known as a point group, which is of finite order, except for those containing the operation CX> , such as exists for the HF or CO2 molecules, for example. g) The properties of the members of a group may be presented conveniently as a group multiplication table (Section 3.2.3), again irrespective of the law of combination. The members of the group are listed in the top row, identity first, and again in the left-hand column. An entry in the top row represents the operation first applied, and that in the left-hand column the second operation; the combination member falls in the appropriate position in the body of the table. The order of operations is important, as indicated in Section 3.2.1, and it is that used normally in mathematical equations. For example, in evaluating exp[sin(x)] we find first sin(x) and then carry out the exponentiation of the resulting value. h) Two groups 6 and 6' are described as isomorphic if there is a one-to-one correspondence between the members of the groups, that is, the combination C = AB in group 6{E, A, B, C} implies the combination C' = A'B' in the isomorphic group 6/{E~ A ~ B~ C'}. i) Two groups 6 and 6' are termed homomorphic if they are similar in structure and have the same group multiplication table. If we associate the groups 6{E, A, B, C} and 6' {l, I, -I, -I} in the following manner, E 1
ABC I -I -1
they show a two-to-one correspondence, because two members of 6 are associated with one member of6': they have the same group multiplication table (see Section 3.2.3), and are homomorphic.
Sec. 3.2]
What is Group Theory
41
3.2.3 Group multiplication tables We examine here three groups that will illustrate and elaborate some of the principles just discussed. a) Consider first an abstract group of order 3 consisting of the members E, A, and B, and for which the relations A 2 = Band AB = E hold under multiplication; the group table may be written as
E
A
B
E
E
A
B
A
A
B
E
B
B
E
A
b) A cyclic group of order 3 comprises the members Z, Z2 (= Z-1) and Z3 (= E) under multiplication; the group table becomes
E
z
E
E
z
z
z
E E
z
c) Finally, consider the group of order 3 comprising positive integers modulo 3, with addition as the law of combination; the group table is as follows:
a
1
2
a
a
1
2
1
1
2
2
2
a
a 1
These three groups are isomorphic. There is a one-to-one correspondence between the members of the groups: A ~Z B ~Z2 C ~Z3
~I
~2 ~O
The groups are also Abelian; this nature is revealed through the symmetry across the principal diagonals of the bodies of the group multiplication tables. We note also that no member of a group is repeated among any row or column within a group table.
42
Group Theory and Point Groups
[eh.3
An important group in studying crystal symmetry is the translation group: the basic translations a, band c that define a unit cell in a three-dimensional lattice comprise an infinite group of the form (3.7)
The law of combination is vector addition, and the zero vector (nl = n: = n3 = 0) represents the identity operation; the negative signs on n, introduce the inverse members of the group. The power of the group multiplication table enables us to write out the tables without necessarily knowing first all the relations between the members. For example, in Problem 1.2(d) we found a C2 axis and two perpendicular mirror planes intersecting in that axis. We denote these planes by cry and o ~, and we can immediately set down a partial group table of the corresponding symmetry operations: a, cr, E C2 E
E
C2
C2
o; a,
C2
cr,
cr,
cr, crv
It is not difficult to enhance it immediately, because of the nature of C2 and o themselves; each operation is its own inverse: E C2 cry crv '
c, or C 2cr, = a v ' . If we choose the former then column 3 within the table would contain o, twice, which is not permitted; hence, we must write C2 crv = o, , whereupon the remainder of the table can be completed: C2v E av av C2 We now have to decide whether C2cr,
=
E
E
C2
cr,
av
C2
C2
E
av
cry
o,
av
crv
E
C2
cr v
av
o;
C2
E
We shall refer to this table again; it relates to the point group C2v (C, 'cyclic'), which is Abelian, order 4. It is of frequent occurrence: for example, water, tetrafluorosulfur, chlorobenzene and the cis-dibromodichloroplatinum (II) anion all share this group.
Sec. 3.2]
What is grout) theory
43
3.2.4 Subgroups and eosets We introduced subgroups briefly under Section 3.2.2, and here we discuss them more fully. The group multiplication table for ~v above indicates three subgroups: the identity group £.{E}, or C1 {E}, the group C2 {E, C 2} and the group C. {E, o}, If we remove all reference to the operation C 2 in ~v, that is, C 2 and one c, because C 2 = o,o ~ , we are left with the operations E and cr. They form the subgroup C. :
C. E
rr
E E
o
c E
There is only one way in which C. may be derived from C2v, for which reason it is called an invariant subgroup of its supergroup C2v . The other subgroups of C2v are also invariant, as shown by Figure 3.1. It does not follow that these subgroups are necessarily invariant with respect to other supergroups, Figure 3.1 (see also Problem 3.4). The order g of a subgroup is always an integral submultiple of the order h of the corresponding supergroup, that is, g = hln, where n is an integer greater than unity. The point group D2h may be written D2h{E , C 2(z), C 2(v), C 2(x ), i, cr(xy), cr(zx), cr(vz)}. One subgroup of D2h is D2 {E, C2 (z), C 2 (v), C 2(x ) } . We can form the product of each member of D2 with a member of D2h that is not also a member of D2, say, cr(xy). If we carry out the right-multiplication of D2 by the set D2cr(xy ) obtained is termed the a right coset of D2 : D2 cr(xy ) = Ecr(xy), C 2 (z)cr(xy ) , C 2 (v)cr(xy), C 2 (x )cr(xy ) = cr(xy),
i, cr(vz), cr(zx).
Thus, the group D2h can be represented by the combination of D2 and the right coset of its subgroup D2 with cr(~v) under addition:
D2h = D2 + D2cr(~V). Left cosets may be constructed in an analogous manner.
44
Group Theory and Point Groups
A~
v
AI' ." 0
[Ch.3
"
~
1
Fig. 3.1 The thirty-two crystallographic point groups and their subgroups; thin lines indicate the subgroups that are invariant. Thus, C2v is an invariant subgroup of D2d, C4v and D2h, but not of C6v or D3h
3.2.5 Symmetry classes and conjugates Subgroups provide one method for separating the members of a group into smaller sets, each constituting a group. An alternative procedure introduces the topic of symmetry classes. In Section 2.5.4, we introduced the similarity transformation; we now use this concept to discuss symmetry classes. If two symmetry operations A and B of a group are linked to a third operation R by the equation B = R1AR,
(3.8)
then B is the similarity transformation of A by R, and A and B are said to be conjugate to each other. Conjugate members of a group have useful properties, as follow: a) Every member of a group is self-conjugate, that is, for any member A A = R1AR:
(3.9)
for left-multiplication with A'I leads to A,IA = E = A,IR1AR = (RAYI(AR),
(3.10)
following Section 2.5.3. It follows that RA = AR so that R and A commute; R may be just E, which is always present, or any other member that commutes with A.
Sec. 3.2]
45
What is grout) theory
b) If A is conjugate to Band B is conjugate to C, then A is conjugate to C. We have
A
R'BR,
(3.11)
B = Q-'CQ.
(3.12)
=
Substituting for B in (3. 11) leads to A = RIQ-1CQR = (QRrlC(QR) = Z-ICZ,
(3.13)
where Z (= QR) is also a member of the group, conjugate to A. c) As a corollary to (b), by putting C = A in (3.11H3.12), it follows that if A is conjugate to B, then B is conjugate to A. We define a symmetry class to comprise those members of a group that are conjugate one to the other. This definition may be illustrated with the aid of the group tables that we have already discussed. We take the cyclic group of order 3, C3{E, Z, Z2}, and perform a similarity transformation of each member by each member of the group, with the following results: E: K1EE = EE = E Z-'EZ = Z-I Z = E (Z2r'EZ2 = (Z2r 1Z2 = E Z: K1ZE = ZE = Z Z-'ZZ = EZ = Z (Z2r IZZ2 = (Z 2rIE = Z Z2: K 1(Z2)E = Z2 Z-I (Z2)Z = Z-I E = Z2 (Z 2r' (Z2)Z2 = EZ2 = Z2 It is evident that this cyclic group contains three symmetry classes, each with one member, E, Z or Z2. It is left as an exercise to the reader to show that the group ~y {E, C2, o, , 0" ~ } contains correctly the four symmetry classes E, C2 O"y and 0" ~ .
We shall content ourselves with one more example group, that exhibited by the trigonal-pyramidal ammonia molecule and symbolized by C3y; its order is 6 and the group multiplication table is given below. We work through one member, C 3, and leave the remainder as an exercise for the reader (see also Problem 3.5). C3y
E
C3
C~
O"y
O"y
o"v
E
E
C3
C~
O"V
O"y
O"y"
C3
C3
C~
E
O"y
O"y"
O"v
2
C~
C :J
E
C3
O"y
O"V
O"y
0",.
"
O"V
O"y
O"y
E
C~
C3
O"y
O"y
0",.
O"y"
C3
E
C~
"
O"y"
O"y
O"y
C~
C3
E
O"y K IC3E = C3 (C3r C 3C3 = EC 3 = C 3 l
(C ~
r'c, C ~
= (C~
rlE
C3
"
O"v-l C 30"v 0" v'I- C30" 'v
O"v_IO" y = O"vO" v =C~
O"~ -IC30"~
0" ~ O"V = C ~
O"yO"y" = C~
46
GrOUI) Theory and Point GroUI)S
[eh.3
From the results just obtained, it follows that the operations C 3 and C:1 belong to one and the same class in this group. In a similar manner, cry, o v and o ~ form a single class, and we write the classes for C3y as E
The orders of classes must be integral submultiples of the order h of the group. In group C3 , the classes are E, C 3 and C 3
:
the operations C 3 and C:1 do not combine
into one class in this group; there is no other operator in the group that will combine with C 3 to give C ~ . In fact, all symmetry classes of Abelian groups comprise single symmetry operators. We need not necessarily engage in the labour of performing these similarity transformations each time we need to determine the symmetry classes of a given group. It will be sufficient to separate those operations that are themselves equivalent under a symmetry operation of the group. Thus, in the group that we have just considered, the three cry operations are equivalent under C 3 , whereas C 3 and C 3 are equivalent under the cry operations: E itself always forms a separate class in all groups; 0- 1EO = E for any chosen operator O.
3.3 DEFINING, DERIVING AND RECOGNIZING POINT GROUPS We have remarked already that Figure 1.4 is an example illustration of group closure, Section 3.2. I: two symmetry operations in a set were combined to produce another operation, also in the set. A collection of symmetry operations for a body is referred to as a point group, which may be defined formally as a set ofsymmetry operations, the action of which leaves at least one point of the body invariant, or unmoved. A similar definition may be expressed in terms of symmetry elements. All symmetry elements of a point group pass through the invariant point. In some cases it is a line or even a plane that remains invariant under the symmetry operations of a point group. The origin of the reference axes must be a point of invariance under all symmetry operations: otherwise, there would arise a multiplicity of origins (and symmetry elements), which is a feature of repeating patterns. Figure 3.2 illustrates the effect of two non-intersecting (parallel) twofold axes; it is clear that they lead to infinite repetition, to patterns that are governed by space-group theory. 3.3.1 Deriving point gmulls The operation of rotation C, (Section 2.2.3) is defined formally for values of n from I to infinity. However, to keep the discussion to a comfortable length, we shall restrict ourselves at first to the thirty-two crystallographic point groups, which means that n can take the values L 2, 3, 4, and 6. Subsequently, we shall consider some extensions ofn, one of which is explicit in Figure 2.2.
Sec. 3.3]
Defining, deriving and recognizing point groups
47
- - - .... 00
+ 1
+
+
00
00
00
2
3
a
5
4
c
b
6
e
d
Fig. 3.2 Effect of two parallel twofold axes (diads) a and b, considered to be lying in the plane of the diagram. Point 2 is related to point I by rotation about axis a; the ± signs indicate a given, fixed distance above or below the plane of the diagram. Points I and 2 rotated about axis b produce points 3 and 4; but 3 and 4 are, themselves, related by another diad, c. The effect of diad c on points I and 2 is to produce points 5 and 6; but they are related to 3 and 4 by diad d, and to each other by diad e. Now 3 and 4 can be rotated about e, and so on. This progression would lead to an infinite number of parallel, equidistant diads, together with the symmetry-related points, an arrangement that is wholly inadmissible under point-group synunetry.
The simplest point group consists of the single symmetry element C\, which corresponds to the identity operation E, or C\: we give this group the symbol C\, and its order h is I. Evidently, we can write another four such simple groups Cn with n = 2,3,4 and 6; we will refer to these groups as type (a). Another five groups are S, (n = 1, 2, 3,4 and 6). We note here that it is conventional to refer to 5\ as C. (it has a single symmetry plane), to 5:l as C, (it has a point of inversion, or centrosymmetry), and to 56 as C3h (it has a C3 axis normal to a symmetry plane); we call these groups type (b). In order to proceed further. it is necessary to consider combinations of symmetry operations: it is evident that combinations such as EC n and ES n cannot lead to any new groups, because E is a member of all groups, so we consider next the combinations iC n .. For n = 1. iC\ = i, which was derived under type (b), as point group Ci (5:l). For n = 2, we have i combined with C 2 . We define a group 6{E, C 2 , i, ...}, and we can construct a partial group table:
E C2
E
C2
E C2
C2 E
i E
It is not difficult to see that the combined effect of rotation by 180 0 and inversion through the centre of symmetry. a point on the C2 axis, is equivalent to reflection, from the initial position, through a symmetry plane normal to the C2 axis. Because of its orientation this plane is designated Gh. The point-group symbol is C2h, and we can now complete its group table:
48
Group Theory and Point Groups
C2h
E
E
E
C2
C2
C2 C2
ah ah
E
all
E
all
ah
ah
[Ch.3
C2
i
i
C2
E
We can show the effect of the combination iC 2 graphically by means of a stereogram (see Appendix 3). In Figure 3.3, the points have been enumerated, and we can trace the following operations:
i ~3
2
~
4
We see that the operation C2 followed by i is equivalent to the reflection operation we write this result as
ah;
(3.14) In this example, the combination C 2i gives an equivalent result; i and C 2 commute. We recognize this equation as another example of closure (Section 3.2.1). This group may be written as C2h {E, C 2, i, all}; h = 4, and its subgroups are C\, C2 , C, and C, . We stress that we symbolize the reflection plane in this example as ah even though it is set vertically with respect to the stereogram: the symmetry plane normal to the principal C, axis is always designated ah, whatever the orientation of the stereogram or other illustration of the group. Conventionally, the z axis (C 2 ) would be set normal to the plane of the stereogram: the variation adopted here allows the four symmetry-related points to be appreciated more readily on the diagram. Proceeding in this manner for other values of Cn, we find that i combined with C, generates all if n is even. When n is 3, i combined with C 3 generates 8 6; the result
Fig. 3.3 Stereogram lor point group C2h; the C2 axis (z) lies lett to right on the diagram, and the O'h plane (x,y), indicated by the heavy line, is normal to C2.
Sec. 3.3]
Defining, deriving and recognizing point groups
49
for n = I has been discussed already. Thus, the combinations of C, and i have led to three new point groups, C:l1" C4h and C6h , denoted as type (c). Next we may consider the combinations is,,. lt is not difficult to show that no point groups are obtained that are not already included under types (b) and (c), and we summarize the thirteen point groups derived so far in Table 3.1. Note the conventional symbols for certain S, groups in type (b); the symbols in parentheses indicate second occurrences of a group within the table. Table 3.1 Type
a) b) c)
Partial set from the 32 crystallographic point groups
Operator/s
Cn Sn
ic,
Number of groups
5 5 3
Point-group symbols n 4 3 2
C1 C, (C i )
C2 C, C2h
C3 C3h (56)
C4 54
C4h
6
C6 56
C6h
Euler's construction In order to derive further point groups, we must combine C, operations with operations other than E or i. Until now, there has been no problem about the relative orientations of the symmetry elements that we have used; i has been a point on an axis. However, if we wish to combine, say, C, and C 2 , we need to know their relative orientations. It seems likely that a C2 axis would lie perpendicular to Cn, or even collinear with it: but are these orientations correct, and are there other possibilites that we should consider? We shall discover that constraints exist that limit the number of possible combinations of symmetry elements. whether or no we are restricting the discussion to just the crystallographic point groups. It is instructive to carry out the necessary analysis by a construction due to Euler. In principle, it is another application of the group law of closure. of which (3. 14) is one example, as we have seen. We may consider Euler's proposition first in terms of proper rotations Cn, and then extend it to include the improper rotations Sn. A geometric illustration of Euler's analysis is illuminating. Let OA and OB (Figure 3.4) represent two symmetry axes intersecting at 0, the centre of a spherical projection of a symmetrical crystal (see Appendix 3); A, Band A' define the equatorial plane, normal to the direction through C, 0 and C'. Let BA C = BA C' = ai2, and ABC = A iJ c = ~i/2, where a. and ~ are angular rotations associated with the axes OA and OB respectively. Consider the motion of the line Ot'. The anticlockwise rotation «, about OA, maps C on to C; the anticlockwise rotation ~1, about OB, returns C to its original position. The sense of rotation given here is for observation along the axis towards the origin O. The arcs CBC and CAe' are zone circles (q.v.)for the axes OA and OB respectively. We can think of C' as the image of C in the plane OAB. lt may help to consider Figure A3.2 (Appendix 3), where OP would map on to OP' by an anticlockwise rotation of 1800 about the axis bOb'; a 90 0 rotation would have taken
50
Group Theory and Point Groups
[Ch.3
P just to the point e in the equatorial plane. Similarly, an anticlockwise rotation of 180° about eOe' returns P to its first location.
A'
Fig 3.4 Partial spherical projection of symmetry directions in a body; OA and OB are the given rotation axes.
The combination of the two rotations, 0.(04) and ~(B), leaves the point C unmoved: consequently, if there is a motion of a point on the sphere arising from the combination of the given two rotations, then the third, resultant symmetry element must pass through C, 0 and C. Consider next the motion of the point A under the same two operations, The rotation 0.(04) leaves point A unmoved: the rotation ~~(B) maps A on to A', where ABC = A' B C = ~/2; A' is, thus, the image of A in the plane 0 Be. In the spherical triangles ABC and A'BC, ABC = A' B C = ~V2, AB = A'B, and BC is common to both triangles. The triangles are congruent, so that A CB = A' CB. Let these angles be y/2; then the anti clockwise rotation y, about OC, maps A, along arc ABA', on to A', Again, we consider Figure A3.2: under rotation o.(A), about bOb', point b is invariant, whereas the rotation ~(B) about eOe' moves b to b', We may write the result symbolically as ~(B)
o.(A) = y(C),
(3,15)
which means that a rotation ex. in one sense about a symmetry axis OA followed by a rotation ~ in the same sense about an intersecting symmetry axis OB is equivalent to a rotation y in the same sense about a mutually intesecting symmetry axis OC (or clockwise about OC), where OA, OB and OC form a right-handed set of axes. We can use triangle ABC to solve for the angles a, band c between the three pairs of symmetry axes OB OC, OC 004, and OA OB; it is the reflection of triangle ABC in the plane AOB. The solution follows the equations for the polar triangle [Appendix 3, (A3.11) and its cyclic permutations], and the relevant portion of Figure 3.4 is shown in Figure 3.5. We know that a (= 24), P (= 2 B) and y (= 2 C) can take only the values 360° (0°), 180°, 120°,90° and 60°, corresponding to n = I, 2,3,4 and 6 in Cn.
Sec. 3.3]
Defining, deriving and recognizing point groups
51
(ii) The value of 360° for 0., p and y is ignored, because it corresponds to the trivial onefold rotation, or identity; (iii) Since we need only the number of combinations of symmetry elements, permutations are ignored so that only those solutions for 0. = 180°, 120°,90° and 60°, with Ps 0. and y S; p, are required;
(iv) Solutions for which one or two of a, band c are zero are ignored, because they correspond to a dimensionality in the problem of less than three.
B
A
P/2
S/2
c
Fig. 3.5 Spherical triangle ABC, to be solved for a, band c
Subject to these conditions, the six nontrivial results listed in Table 3.2 are derived. In interpreting this table, we recall from Euler's construction that 0., ~ and y refer, in order, to the anticlockwise rotations about the symmetry axes OA, OB and OC', with a, b and c being the angles between the pairs of axes OB OC', OC' OA and OA OB, respectively. Thus, in type (e), for example, we have two C2 axes (0., P) with a C4 axis (y) normal to them, the angle c between the C2 axes (OA and OB) being 45°. Figure 3.6 illustrates the relative orientations of the symmetry axes, and the stereograms show the effect of the symmetry operations on a typical point or vector in an object ofthe corresponding symmetry. In the stereograms the vertical direction (the normal to the stereogram) is the z axis and, except for Figure 3.6(h), corresponds to the rotation axis of highest degree in the group. The y axis is the left to right direction, and the x axis is perpendicular to both y and z. In Figure 3.6(h), the x, y and z axes are C2 symmetry axes, with C3 axes along the body-diagonal directions of a cube; in Figure 3.6(i), the x, y and z axes are each C4 symmetry axes with C3 axes again along the cube diagonals. Where are the C2 axes in this example? In those groups for which there is a single axis of threefold or sixfold symmetry, the orthogonal reference axes do not all coincide with the symmetry axes. However, in studying crystal symmetry, it is conventional that the crystallographic reference axes are taken along the directions of the symmetry axes, and we shall consider this situation more fully in Chapter 8.
52
[eh.3
GrOU)l Theory and Point Groups
Table 3.2 Results following from Euler's construction Type
«/deg
Wdeg
yldeg
aldeg
bldeg
c1deg
d)
180
180
180
90
90
90
e)
180
180
90
90
90
45
f)
180
180
120
90
90
60
g)
180
180
60
90
90
30
h)
120
120
180
cos" (11...)3)
cos" (1/...)3)
i)
90
180
120
cos"(...)6/3)
cos" (1/...)3)
cos" (1/3) 45
3.3.2 Building up the point groups Now that we have determined the angles at which C, symmetry axes may intersect, we can list the point groups corresponding to the types given in Table 3.2. For types (d) to (i) they are, in order D2 , D4 , D3, D6 (D, dihedral), T (tetrahedral), 0 (octahedral). We explore now the combinations of C2 and S!, but before we commence this analysis we note the following general combinations, which may be deduced readily either by application of (3.1) under multiplication, or with stereograms: (Proper rotation) (Proper rotation) = Proper rotation (Proper rotation) (Improper rotation) Improper rotation (Improper rotation) (Improper rotation) = Proper rotation
=
For type (d) in Table 3.2, the combinations of mutually perpendicular Cn and 8 1 axes listed in Table 3.3 are permitted. The first entry in Table 3.3 lists three mutually perpendicular twofold axes, and the point-group symbol for this combination is D2 , which we have already noted (Figure 3.6). Next we replace two (Why not just one?) C 2 axes by S\ axes. We have seen that 8 1 is equivalent to a a reflection plane normal to the direction of the SJ axis, and we have actually considered this combination already in Section 3.2.3; it corresponds to point group C2v. No other combinations are possible for type (d). However, we must always consider if we have formed a group in each type that contains a centre of symmetry, that is, it includes the operator i. Neither D2 nor C2v is centrosymmetric, so now we combine the operation i with D2 , giving the partial group D2h {E, C 2 (z), C 2 (y), C 2 (x), i, ...}. We know already that iC 2 is equivalent to a a plane normal to C2 : thus, we have further the operators a(xy), a(zx) and a(vz) perpendicular to C 2 (z), C 2 (y) and C 2(x), respectively. No new operators are produced by any other combinations within this group: thus we obtain the character table below. We see that this point group is Abelian (all pairs of symmetry operators commute) and its order is 8. Its symbol, D2h, shows that it can be obtained from D:! by combining a reflection plane normal to a C2 axis in D:!. This result may be confirmed by comparing the stereograms for D2 and D2h in Figure 3.7. The combination of i (or ah) with the
Sec. 3.3]
Defining, deriving and recognizing point groups
53
45°
+0
-0
d
e
f
h
Fig. 3.6 The permitted angles between rotation axes C;
9
54
[Ch.3
GroUI) theory and point groups
Table 3.3 Permitted combinations of C2 and SI for type (d) OA (x) axis
OB (y) axis
C2
OC' (z) axis
C2
s,
C2 C2
s,
D2h
E
C 2 (z)
C 2 (y)
C 2 (x )
E C2 (z)
E C2 (z)
C 2 (z) E C 2(x )
C 2(x>: C 2 (y): C2 (y):
C2 (x)
C2 (x)
C 2 (y) C2 (x) E C2 (z)
cr(vz)
o'(zx) cr(xy)
C2 (y)
cr(xy) cr(zx) cr(yz)
C2 (y)
................ cr(xy) cr(zx) cr(yz)
C2 (y) ~(~yj
cr(vz) cr(zx)
E
.. .'cr(zx) ""(j-(Vi)' cr(xy)
Point-group symbol
cr(xy) cr(zx) o (vz)
E C2 (z)
C2 (y)
C2 (x)
D2
C2v
cr(xy)
cr(zx)
cr(yz)
cr(xy)
cr(zx) cr(Yz)
cr(Yz) cr(zx) cr(xy)
o'(pz)
cr(zx) C2 (z) E C2 (x)
C2(y)
cr(xy)
C2 (y)
C2 (x)
C2 (x) E C2 (z)
C2 (z) E
C2 (y)
symmetry elements of D2 introduces all remaining symmetry elements of D2h . If we repeat this exercise with C2v• we find that its combination with i (or crh) again produces D2h . These results may be summarized by the scheme +i or rr.. ~
Direct products of groups We may express these relationships neatly in terms of a direct product of groups, either D2 or C2v with Cj . The group multiplication table for D2h (above) may be
m \±] (a)
(b)
Fig. 3.7 Stereograms tor point groups (a) D2, and (b) D2h: the introduction of the centre of symmetry into D21eads to mirror planes perpendicular to all C2 axes ofD2.
divided into quadrants, that outlined in the table being the group multiplication table for D2 . A direct product of D2 and Cj is obtained by forming all possible products between the symmetry operations of the two groups, which we may write as D2 (8) C, = D2h or, more conveniently
(3.16)
Sec. 3.3]
Defining Point Groups
55
where the direct product symbol 0 implies all possible products under multiplication between the symmetry operations of O2 and i. It is not difficult to see that the following direct products also lead to point group 02h: O2 0 ah = C2V 0 i = C2v 0 ah =
02h
(3.17)
02h 02h
C 2 in the third equation (3.17) acts in a direction normal to ah (see also Section 3.2.4). We consider one more example of point-group derivation from the data in Table 3.2, namely type (f), The axes OB and OC' correspond to the reference axes y and z respectively, whereas the x axis lies between two C2 axes, as shown in Figure 3.8a. Although only three symmetry axes were needed in Euler's construction, a C2 axis normal to a Cn axis leads to a total of n C2 axes that are equivalent under the symmetry C; Table 3.4 indicates the permitted combinations of one C3 axis with C2 and SI axes. In point group 0 3 , there are three C2 axes in the xy plane, normal to the C3 axis (z) and related by it. The group may be written formally as 0 3 {E, C3 , C:i , C 2, C 2" Cs-}; its order is six, and the group multiplication table is easily constructed. The stereogram in Figure 3.8a illustrates this point group; it appears also among the six unique solutions in Figure 3.6. If we replace the C2 axes by SI axes then, because the Sl axes are normal to the C3 axes, the symmetry planes to which they are equivalent are vertical (3a v) and intersect in the C3 axis. Thus, we have the group C3v {E, C 3 • C:3 , a v , a v , a ~ }:
its order is 6, and its group table has been given in Section 3.2.5. Each of the point groups O2 and C3v combines with i to form D3d : +i (-
The stereograms in Figure 3.8 illustrate these point groups and their interrelationships. In this way we may continue to derive the remainder of the thirty-two crystallographic point groups: we would find that no roto-reflection axis other than SI is needed in this derivation; the other permitted S; axes emerge during the Table 3.4 Point groups derived from type (f) OA axis
OB (v) axis
OC' (z) axis
Point -group symbol
Group theory and point groups
56
[eh.3
--> OY • Y axis
1
X
ax t e
(b) Fig. 3.8 Stereograms to show point groups (a) D3, (b) C3vand D3d; D3d may be formed from D3 or C3v by direct product with Cj ••
derivation. A summary of the crystallographic point groups, together with the two groups for cylindrical symmetry (linear groups), is provided in Table 3.5. A program, EULS (EULH in the Hermann-Mauguin notation), is available on the Internet Web site www.horwood.net/publish (from where further user directions are obtainable) that leads the user through the various stages of Euler's construction as discussed above; it considers also certain noncrystallographic point-group symmetries. The reader is encouraged to make use of this facility. Table 3.5 Point groups and their symmetry operations Group
Operations
C1
E
Cz
E,
C3
E, C3"'C~
C4
E, C 4
CZ
Cs
E,Cs
C;
C6
E, C 6
C 3 ••• C Z... C ~ ... C ~
Cn
:
One n-fold (proper) rotation axis
c, C~ C~ ...
ct
C.: One mirror symmetry plane (a)
c,
E, a Cj
:
One point of inversion (centrosymmetry)
Sec. 3.3]
c.
Defining Point Groups
57
E, i S, : One n-fold roto-reflection axis (n even)
54
E, S4 .. ,S2."S ~
56
E, C 3 ... C i
... 8 6 ... 8 ~
,i
Cnv : One C, axis and n vertical c, planes intersecting in C n
C3v
E, C2 , 20 v E, C3 C i , 30v
C4v
E, C 4
Csv
E, 2C s
C6v
E, C 6 C 3... C 2 ... C ~ , C ~ , 30v , 30d
C2v
C 2 ... C ~ , 20v , 20 d
2C; , 50v Cnh : One C, axis and one 0h plane normal to C«
C3h
E, C 2 , i, 0h E, C3 C~
C4h
E, C 4
C2
CSh
E, C,
C;
C6h
E, C 6 C 3 C 2 ... C 3
~h
ah
83"'S~, C~
8 4 ... 5 ~ . i, 0h
C ~ ... S5
C~
... C 6
5 ~ ... 8; 8 6 ... 8 3 S 3
8~. , 0h ... 8 6 ,
l, 0h
On: One C, axis and n C2 axes normal to it
E,3C 2 E, C3"'C~,
E, C 4
3C~ C 2 ... C ~ , 2C ~ , 2C ~
E, 2C 5 2C ~ ... 5C ~ 5 E, C 6... C 3... C 2 ... C "'3 ... C 6' 3C •2, 3C "2
0nh: As On ' plus one (horizontal) 0h plane normal to C;
E, C 2 (z), C 2 (y ), C 2 (x ), i, o(xy), o(zx), o(yz) E, C 3 C ~ E, C 4
C2
S3 S ~ , 3C C~
'2'
ah, 30 v
S4... S ~ , 2C ~ , 2C ~ , i, 0h, 20 v , 20 d
E, 2C s 2C ~ ... 2S ... 2S ~ ... 5S ~ , 0h, 50v E, C 6... C 3... C 2 ... C 30d
2 5 "S'" • 3 ... C 6 ... 5 6 ... 5 3...5'3 ... 5 6 , 3C 2' 3C 2' I,
0h, 30v ,
Ond : As On, plus n vertical (dihedral) ad planes intersecting in Cn
Group theory and point groups
58
1
[eh.3
'
E, C Z••• S4 S 4, 2C 2' 2ad
E, C 3 .•. ci
,3C'2 .i, 3ad
S6"'S~
E, 2S8oo .2C 4°o .s.. .2S R , 4C ~ ,4ad
Cubic groups: T, tetrahedral; 0, octahedral. T
r, r, o 0h
E, 4(C 3 • ooC i), 3C z
i ",S6°o' S ~), 3C z. i, 3ah E, 4(C C i), 3(C ooS4°o'S 1)6 a d E, 4(C 3°o'C D, 3(C 4oo.C zoo. C 1). 6C 2 E, 4(C 3°o'C i oo,S6°o'S 6), 3(C C 1.·S4oo ,S 1). 6C 'z, i, 3ah, 6ad E, 4(C 3.ooC 3°o'
2.
4oo.CZoo.
Coov :
One C" axis and an infinity of a v planes intersecting in
Dooh : As C oov, plus one ah, plane normal to Dooh
E, (C~
.ooC ~vl ... S ~ ... S ~vl),
Coo
Coo
ooC ~ ,i, ooav
Notes on Table 3.5
1) Symmetry operators linked by the dotted line ... act collinearly. 2) A vertical symmetry plane is designated ad if it bisects the angle between twofold axes that lie normal to the principal C n axis. In point groups C4v and C6v , the labelling of the ad planes is not strictly consistent with this rule, but the notation is general [lZ l . 3) In point group D4 , the symmetry element C2 ' lies along the x axis and C2 " between x and y; in D6 , C ~ is along y. 4) In point groups D4h and D6" . the o, planes contain the symmetry elements 2C ~ ,
5) 6) 7) 8) 9)
and ad the elements 2C'~ . Strictly. all planes are ad type but the conventional notation is used here. In the linear point groups Coov and D",h, ~ represents an arbitrary angle of rotation. The order of symmetry operations listed here will not necessarily be the same as that in the corresponding character tables. The point-group symbols C. and C, are preferred over S, and Sz. Of the five cubic point groups, Td and 0" are of particular significance in chemistry. The unprimed symbol C z refers to an operation along the z axis, or the x, y and z axes in the cubic point groups.
Sec. 3.3]
Defining Point Groups
59
10) The symbols C 2, and C 2" refer to different forms (q.v.) of C 2 operations, n of each, that are symmetry-related under the principal Cn operation. The symbols C 2, C 2" C 2 " are used to indicate the n C 2 axes in one and the same form. II) The parentheses in the cubic groups include operators with a common multiplying factor, but which are not necessarily of the same symmetry class. Thus, in r, 4(C 3 •.. C ~ ) leads to the class 8C 3 , whereas in Oh 4(C 3°o'C ~ .S6... S ~ ) leads to the classes 8C 3 and 8S6. 00
3.3.3 Federov and Plato solids Federov showed[62j that there are only five polyhedra, each type of which can be packed in one and the same orientation to fill space completely. Figure 3.9 illustrates the Federov solids; they are also Wigner-Seitz cells (q.v.), derived from Bravais lattice unit cells: Federov solid
Crystal form/s
Lattice unit cell
Cube Rhombic dodecahedron Cube + octahedron Hexagonal prism Elongated rhombic dodecahedron
{100}
Cubic? Cubic I CubicF Hexagonal Rhex Tetragonal I
{llD}
{100} + {Ill} {l0 1 O} {l00} + {lII}
If the program EULS (or EULR) has been used in studying Euler's construction, it would have been noticed that, although the program extends the discussion given here to include the noncrystallographic rotation axes Cs, C7 and Cg, no point groups emerged for which there were intersecting rotation axes of degree greater than 5, as in point group I, for example. This result is related to the existence of only five regular solids, the Plato solids, that is, those with regular polygonal faces, and among which the maximum proper rotation axis found is Cs: a regular polygon has equal sides and equal angles between successive sides.
(a)
(b)
(c)
(d)
(e)
Fig. 3.9 Federov solids: (a) cube, (b) rhombic dodecahedron, (c) cube + octahedron, (d) hexagonal prism, and (e) elongated rhombic dodecahedron. The cube is also a regular, Plato solid.
The Plato solids may be determined from the following argument. Consider a regular polygon with p sides, and let q such polygons form the comer of a polyhedron. Since the internal angle of a poly-p-gon is [90(2p - 4)/p]0 and the sum of the angles formed by the faces at any comer must be less that 360°, it follows that q(2p - 4)/p < 360/90,
(3.18)
GrOUI) theory anti point groups
60
[eh.3
which simplifies to
l/q + lip> \12.
(3.19)
Euler showedl13] that the numbers of faces f, edges e and corners c of a polyhedron are related by the equation c + f= e + 2.
(3.20)
Since, p and q must each be greater than 2, the only values of p and q that satisfy inequality (3.19) are those listed in Table 3.6. The less familiar pentagonal dodecahedron and the icosahedron are shown in Figure 3.10, and both exhibit Cs and S\o symmetry axes.
(a)
(b)
Fig. 3.10 (a) Pentagonal dodecahedron, (b) Icosahedron
Table 3.6 Values for p and q in (3.19), and the Plato solids p,q 3,3 3,4 4,3 3,5
e
5,3
30
6 12 12 30
f
c
4 6 8 12
4 8 6 20
20
12
Plato solid Tetrahedron Cube Octahedron Pentagonal dodecahedron Icosahedron
Point group
Td
°h °hI h
Ih
These two solids are related to each other in the same way as are the cube and octahedron, that is, by an interchange of the numbers of faces f and corners c. The 3,5 and 5,3 solids have the same symmetry, and their analyses are summarized in Table 3.7. From the symmetry axis SIO there are the operations E, 510 , 5 ~o
c ~ , 5 io =
=
C s, 5 fo , 5
to =
= i, S fo = 5; , S io , S fa = C ~ and S io ; from Cs we obtain E and C 5 (n
1-4), all included with the SIO operations. From S6 we obtain the symmetry operations E, 56, 5 ~
= C3 ,
S ~ =i, S:
=C~
and S ~ : under C3 we have the operations E, C 3 and C ~ , which are included with the S6 operations. There are also fifteen C ~ operations and fifteen a operations in each solid.
Sec. 3.3]
Defining Point Groups
61
Table 3.7 Analysis of the pentagonal dodecahedron and icosahedron Symmetry element
Orientation of symmetry element Pentagonal dodecahedron Icosahedron
Six SIO through:
centres of opposite pairs of pentagonal faces
opposite vertices
Six Cs collinear: Ten S6 through:
with SIO opposite vertices
with S,
Ten C3 collinear: Fifteen C 2 bisecting:
with S6 pairs of opposite edges
with S6 pairs of opposite edges
Fifteen cr containing:
two C s and two C2
two C s and two C2
centres of opposite trianglar faces
If we analyse these data in the customary manner, we would deduce the following symmetry classes for the pentagonal dodecahedron and the icosahedron: E, 12C s, 12C; , 20C 3 , 15C ~ , i, 12S10 , l25?o, 205 6 , 15cr. The point group is known as Ih and its order is 120. If we excise all reference to the centre of symmetry, that is, all 8 10,56 , cr and i, we obtain the classes E. l2C s, 12C ~ , 20C 3 , 15C ~ , which relate to the pure rotation subgroup I. We shall not meet examples of point groups I and Ih in our study, but they are encountered with some molecules of biological interest. 3.3.4 Practical recognition of point groups In the scheme for practical point-group recognition to be described here, molecules and crystals are divided first into four groups depending upon the presence of a centre of symmetry, or a mirror plane, or a centre of symmetry and a mirror plane, or neither of these two symmetry elements, leading to Table 3.8. The recognition of a centre of symmetry or of a mirror plane is very straightforward: for a centre of symmetry, place the given model on a flat surface; then, if the plane through the uppermost atoms (in the case of a molecular model) or the uppermost face of the crystal (in the case of a crystal model) is parallel to the supporting surface, a centre of symmetry is present in the model. If a mirror plane is present, it divides the model, conceptually, into halves that are related as an object is to its mirror image, the well known right-hand-s-left-hand enantiomorphism. It is suggested that the reader examine these simple rules with models of SF6 (or a cube) and CH4 (or a tetrahedron), or other examples. A correct identification at this stage is important: it places the model into one of types I-IV, shown by Table 3.8 and Figure 3.11. Further study concentrates on the nature of the principal symmetry axis, the number of them, if more than one, the presence of other mirror planes or twofold axes and so on, according to the scheme illustrated in Figure 3.11. It will be apparent that this scheme resembles a flow diagram of a computer program'!". Indeed, the procedure can be carried out with the program SYMS which is accessible
62
Group Theory and Point Groups
[eh.3
Further study concentrates on the nature of the principal symmetry axis, the number of them, if more than one, the presence of other mirror planes or twofold axes and so on, according to the scheme illustrated in Figure 3.11. It will be apparent that this scheme resembles a flow diagram of a computer programl'"'. Indeed, the procedure can be carried out with the program SYMS which is accessible
tIl
~ ~ n
>
17
,o
(II)
"~
1.------+1
C
~
YLa'" .nd
'.'"
Yo.
l~
n
-
'="
9'O!~----101
I
4.7
n
No
I-I
-----+i T ,
~
)0
11
C_I
~
r ee
I ... I'
'I
I
i I
I I
I
ur n
' 1? -~--{~
~
56
II
~1r1~:o.Lcn?i~~
I
D
ae
I
! I I I
Fig.3.11 Block diagram of the point-group recognition scheme embodied in the program SYMS (and SYM the scheme under types (Il) and (IV), respectively.
64
GrOUI) theory and point groups
[eh.3
Table 3.8 Point-group recognition scheme: the basic divisions Type I Type II Type III Type IV C1 C2 c, C2v c, 56 C2h C4h C3
C4
C3v
~
~
~
~
~
~
T
0
Caw
D4
D6
D2d
C4v
C6h ~
~
~
~
~
r,
r,
o,
Dooh
under the WWW reference already quoted; a key to the link between a 'model number' and its point group is also provided. For best results the model should have been studied carefully along the lines indicated'" before using the program. If an error is made in the deduction process, the program returns the user to the point in the scheme at which the error has been made. Up to two such returns are allowed before the program offers a suggestion for alternative action. The thirty-two crystallographic point groups are illustrated in Figure 3.12, using the traditional stereogram notation. The two diagrams for each point group show the symmetry elements, and the general equivalent positions that are related by the point-group symmetry. General positions do not lie on any symmetry element: special equivalent positions exist on symmetry elements, and the number of them for a given point group is a submultiple of the number of general positions. For the point groups C2 , C, and C2h , the two diagrams given relate to the choice of the twofold axis: it may be taken along z, which is conventional in molecular symmetry, or along y, which is the universal crystallographic convention. Table 3.9 provides a comparison of point-group symbols in the Schonflies and Hermann-Mauguin notations, with example compounds for each group.
Defining, deriving and recognizing point groups
Sec 3.3]
65
0 CD8 ffiEB
00 ffiEB CD CD8 ffiEB i \ ffi ,:r, CDi CDS C1
(l)
C2
(2)
C4
(4)
(m(=2))
84
(4)
(2im)
C
-
c,
<
(1)
C
2h
1-1-1
C2
(2)
Cs
(m)
a •
"'-1/
a •
4h
(4Im)
• • a
a
_-¥-t VI~I ..........
CDCD CDEB ffi® -
D
C
2
2u
(222)
(mm2)
D4
(422)
C
(4mm)
4v
ffi~'
-
€B a a. • -
wEB C 2 h (21m)
D
CDri\ t-t-t \....1/
00 D 2 h (mmm)
o o
2d
00
\(IY
(42m)
0
0
I
~
11.. I I" ~.(~,--~ \
I~
00 .... ~ D 4 h (4Immm)
Fig. 3.12 Stereograms of the 32 crystallographic point groups: the 1st and 2nd settings for monoclinic crystals correspond to the unique, twofold axis along z and y, respectively. Both symmetry notations (Table 3.9) are used in this figure.
, I
I
i
66
Group theory and point groups
.® ·.-J-. \ j \,J I '\- ~
.. @ . ®@ o.
D3
0
(32)
.. •
•
••
-
•
•
:. .: ••
c6 u (6mml
[Ch.3
Defining, deriving and recognizing point groups
Sec. 3.3] Table 3.9
Synunetry notations and example compounds
Schonflies Hermann-Mauguin
Cj ~
C3 C4 C6 C, C. S4 S6 C2h C3h C4h C6h C2v C3v C4v C6v O2 03 04 06 02h 03h
Example or possible example
I 2 3 4
CHBrCIF, bromochlorofluoromethane H20 2, hydrogen peroxide H 3P04, phosphoric acid (CH3)4C4, tetramethylcyclobutadiene
6 -
C6(CH3) ~ , pentamethylcyclopentadienyl
I
m
-
C6HsCH2CH2C6Hs, dibenzyl C6H3Ch, 1,3,5-trichlorobenzene
4
[H 2P04L dihydrogen phosphate ion
3
[Ni(N0 2)6( , hexanitronickelate(II) ion CHCl=CHC I, trans-I,2-dichloroethene
-
21m 6
41m 61m mm2 3m 4mm 6mm 222 32 422 622 mmm 6m2 4
C 3H3N3(N3h, 1,3,5-triazidotriazine [Ni(CN)4f, tetracyanonickelate(II) ion C6(CH3k hexamethylbenzene C~sCI, chlorobenzene CHCh, trichloromethane [SbFsf , pentafluoroantimonate(V) ion C 6(CH2Cl)6, hexachloromethylbenzene C8H 14, cycloocta-I,5-diene [S206f, dithionate ion Co(H 20)4Cb, tetraaquodichlorocobalt C6(NH2)6, hexaminobenzene C6H4C12, 1,2-dichlorobenzene [C0 3f, carbonate ion
04h
-mm
[Aufira]", tetrabromoaurate(III) ion
06h
-mm
6
C6H6 , benzene
02d
42m 3m 23 m3 43m 432 m3m
ThBr4, thorium tetrabromide
03d T Th Td 0
Oh
In
m
C 6H12, chair cyclo-hexane C(CH 3)4, 2,2-dimethlypropane [CO(N0 2)6f , hexanitrocobaltate(III) ion CH 4, methane C8(CH3 ) 8, octamethlylcubane SF6, sulfur hexafluoride
Noncrystallographic point groups
67
68
Group theory and point groups D4d
82m
Cs
5
DSh
5m 10m2 5m
c; DSd
c., Droh
om
odm
[Ch.3
8 8 , sulfur CS(CH3 ) ; , pentamethylcyclopentadienyl CsHsNiNO, nitrosylcyclopentadienylnickel (CsHshRu, biscyclopentadienylruthenium (CsHshFe, biscyclopentadienyliron HCI, hydrogen chloride CO2 , carbon dioxide
PROBLEMS 3 3.1 Which pairs of operations C 2 , C 3, rr and E commute? 3.2 Construct a group multiplication table for (a) C2h and (b) D2d . Is either of these groups Abelian? 3.3 Construct a multiplication table for the cyclic group 6{E, A, B, C} of order 4, using the relations A = r, B = ?, C = r 3 . Which operation, other than E, is its own inverse? 3.4 Figure P3.1 is a stereoview of the dithionate ion, as seen along the s--s bond. (a) Determine the point group of this species. (b) Construct a group multiplication table for this point group. (c) Determine the subgoups of this point group. (d) Which, if any, ofthe subgroups is invariant to the point group of the species. 3.5 By means of similarity transformations, or otherwise, show that the three cr operations in C3v fall into one symmetry class.
Fig. P3.1 Stereoview of the dithionate ion, along the S-S bond.
3.6 From a consideration of group multiplications tables, show that the point group
54 cannot be obtained from any combination of symmetry operations, whereas 56is equivalent to a certain combination. Then, determine this combination. 3.7 What point groups are formed by the direct products of (a) D4 and i, (b) C4 and cr(xy), and (c) i and cr. 3.8 What point groups may be developed from type (e) in Table 3.2'1 3.9 Use stereograms to show that C4v combined with i leads to D4h . 3.10 Determine the point groups of each of the following chemical species from their stereoviews:
69
Problems
a
b
c
70
Group Theory and Point Groups
[eh.3
d
e
f Fig P3.2 Stereoviews of the structures of six chemical species: (a) l,3 ,5-triazidotriazine; (b) chlorobenzene (c) cyclohexane (chair); (d) methane; (e) octafluorotantalate(V) ion (square antiprism); (f) hexachloroplatinate(IV) ion.
Problems
71
3.11 Show that a group consisting of the unique members E, A, B, C, D and F cannot have any member more than once in any column or row within the table. 3.12 Show that if any two members P and Q of a group satisfy the relation P = then P and Q commute. 3.13 How many groups may be constructed, and what are their types, for (a) order 4, and (b) order 5? Draw up group multiplication tables and list any members that are their own inverses. 3.14 Which of the symmetry operations of the point group of the dithionate ion (problem 3.4) are conjugate one with the other? 3.15 Is it true or false that the following pairs of symmetry operations commute? (a) C,... Cn~ (b) C, and i; (c) cr(xy) and cr(vz),; (d) C2 (z) and C2(y); (e) C, and crh; (f) C 4 (z) and C 2 (x ); (g) S, (n even) and i. 3.16 Comment on the equation crb C 4 = 8 4 in its relation to 54, the point group for C~, or a regular tetrahedron. 3.17 Use the point-group symbol D4h to show the power of the Schonflies notation. Take the crh plane normal to the C4 axis and the C ~ axes in the c plane (along x and y), all indicated by the symbol, and show by means of a stereogram, or otherwise, that all other symmetry elements of the group are introduced. 3.18 Show that the members ma (m = -CXJ, ... -1, 0, 1, ...CXJ) form a group under a certain law of combination, and state that law. 3.19 This problem and the next are designed for use with the program SYMS (or SYMH). The numbers in parentheses are the model numbers for use with the program. Determine the point group for (a) a regular tetrahedron (23), (b) a tetragonal pyramid (87), and (c) a hexagonal prism (33). Give the results in both the Schonflies and the Hermann-Mauguin notations. 3.20 Determine the point group for the species (a) cyclohexane in the chair form (40), (b) 1,2,4-trifluorobenzene (71), and (c) trichlordeuteromethane (42). Give the results in both notations. 3.21 Draw four irregular but identical quadrilaterals on a thin card. Cut them out and arrange them to form a symmetrical figure. What is the point-group symmetry of the resultant figure? 3.22 Figure P3.3 can be used to construct a model of a rhombic dodecahedron, a Federov solid. Make an enlarged photocopy of the figure; a second sheet of A4 paper glued to the back of the photocopy adds a useful rigidity. Cut out the figure, lightly score the dotted hinge-lines and fold all portions in the same sense. With a quick-drying glue, attach the flaps to the faces, flap A to face A, and so on. What is the point group of the model? If you are using the program SYMS (or SYMH) for point-group recognition, this model can be numbered 10.
o'ro.
GroUI) theory and point groups
72
B
I I I I I I I
-,
,,
E
,,
,,
......
,
I
D
,
-,
~
~:~
I
I
I
I
I
I
I
I
I
~II
.:~' , ,
,
F
,
,,
,,
,
">4
H
I
I
J
I
: _-- " [::> I
,,
, , ,
. - -?'
:
-,
\"('"
--_..'.
I
',- -~'
,
""'~
/'" j'" "
I
I
I
(B:/
Fig. P3.3 Model of a rhombic dodecahedron
A
\
-, -,
c
[eh.3
3.1 INTRODUCTION: GROUPS AND GROUP THEORY In this chapter we shall discuss the topics of group theory and point groups. We may look upon group theory as a tool for handling symmetry operations and their combinations in a quantitative manner, leading to an elegant procedure for describing the role of symmetry in chemistry. A point group provides a concise description of the symmetry of a finite body, and all finite bodies may be classified in terms of their point-group symmetry. The set of symmetry operations comprising a point group constitutes a mathematical group. 3.2 WHAT IS GROUP THEORY Group theory deals with sets of operations having the property that, when two operations of the set are carried out successively, the result is that which would be obtained by another single operation of the set, starting from one and the same initial situation. A set of operations forms a group provided that it satisfies this condition together with a number of group postulates. 3.2.1 Group postulates A group consists of a set of mathematical objects that may be symbolized by E, A, B, C,..., called members (also known as 'elements') of the group. In our discussion of group theory the members of the group will be, normally, symmetry operations, and they will be written in bold type. Thus, a symmetry group may be written collectively as 6{E, A, B, C, ...}, and all groups are governed by the following group postulates. Closure The combination of any two members of a set is also a member of the set; this group property is referred to as closure, and is implicit in Figure 1.4. The combination of the members is termed their product irrespective of the nature of the combination process; it will generally, but not necessarily, be multiplication. We may write a product as BA=C, (3.1)
Sec. 3.2]
What is Group Theory
39
where C is a member of the set A, B, ..., and we need to indicate the law of combination. If the law of combination is multiplication, then the group is said to be closed with respect to multiplication. Laws of combination In the case that the law of combination is multiplication, we have shown in Section 2.5.3 that the order of multiplication, normally, is important: if (3.1) represents a combination by multiplication then, in general, the reverse order of multiplication leads to a different result: AB=D,
(3.2)
but where D is also a member of the set A, B, C, .... Other laws of combination are inter alia addition, matrix multiplication and vector addition. Association The associative law (see also Section 2.2) holds for multiplication, that is, A(BC) = (AB)C.
(3.3)
IfBC = Z, and with (3.2), it follows that AZ= DC.
(3.4)
Identity member The group contains the identity, or unit, member E (see also Section 2.2.7), such that (3.5) AE = EA = A for each member of the set. Inverse member Each member of the set has an inverse (see also Section 2.5.3) that is also a member of the set; thus, (3.6) We see that these postulates are paralleled by the operations of matrices: this property will be taken up in later chapters, where we can introduce notation to distinguish between a symmetry operation and the matrix used to represent its action. 3.2.2 General group definitions a) A group may be finite, the number of members of the group being the order h of the group, or infinite. Thus, 6{E, A}, A 2 } is finite, but 6{E, AI, A 2 , ... , oo} is infinite; infinite groups have an important role in space-group theory. b) An abstract group is concerned only with the relationships in the set of operations, there being no specific interpretation attached to the members of the group. Thus, 6{E, A}, A 2 } could be a finite, abstract group, since no meaning is attached to the members other than E. c) A group in which all the members commute (see Section 2.5.3) is called an Abelian, or commutative, group. Thus, the group 6{E, AI, A 2 } , where EA i =
40
Group Theory and Point Groups
[Ch.3
A;E (=A;) andA;Aj=A;1; (i,j = 1, 2), is an example ofa commutative group. It is possible for a group to be nonAbelian although some of its members commute; point group D4., which we shall meet later, is one example. d) A group consisting ofa single member A and its powers zl' , A 3 , •.. , AP, is called a cyclic group of order p, where p is the smallest integer for which AP = E. Figure 2.1 illustrated the cyclic group C4 of order 4, containing the symmetry operators
C 4, C ~ (= C 2 ) , C ~ (= C 41 ), C: (= E). We see that operating with C 4 twice in
succession takes us from (a) to (c) via (b), and that operating with C 2 on the initial state takes us from (a) to (c) directly; thus, C2 is properly a member of the group C4 . We note also that the symmetry element C4 is associated with more than one symmetry operation, as discussed already in Section 2.2. e) A subset of members of a given group forming another group is a subgroup of the given group; the given group may, in turn, be termed a supergroup of the subgroup. Thus, a group consisting of the members C 2 (= C 21 ) and C ~ (E), is a cyclic group of order 2, C2 , and a subgroup of the group C4 which, itself, is a supergroup of C2 . f) Every geometrical body, crystal or molecule in our applications, may be characterized by a group known as a point group, which is of finite order, except for those containing the operation CX> , such as exists for the HF or CO2 molecules, for example. g) The properties of the members of a group may be presented conveniently as a group multiplication table (Section 3.2.3), again irrespective of the law of combination. The members of the group are listed in the top row, identity first, and again in the left-hand column. An entry in the top row represents the operation first applied, and that in the left-hand column the second operation; the combination member falls in the appropriate position in the body of the table. The order of operations is important, as indicated in Section 3.2.1, and it is that used normally in mathematical equations. For example, in evaluating exp[sin(x)] we find first sin(x) and then carry out the exponentiation of the resulting value. h) Two groups 6 and 6' are described as isomorphic if there is a one-to-one correspondence between the members of the groups, that is, the combination C = AB in group 6{E, A, B, C} implies the combination C' = A'B' in the isomorphic group 6/{E~ A ~ B~ C'}. i) Two groups 6 and 6' are termed homomorphic if they are similar in structure and have the same group multiplication table. If we associate the groups 6{E, A, B, C} and 6' {l, I, -I, -I} in the following manner, E 1
ABC I -I -1
they show a two-to-one correspondence, because two members of 6 are associated with one member of6': they have the same group multiplication table (see Section 3.2.3), and are homomorphic.
Sec. 3.2]
What is Group Theory
41
3.2.3 Group multiplication tables We examine here three groups that will illustrate and elaborate some of the principles just discussed. a) Consider first an abstract group of order 3 consisting of the members E, A, and B, and for which the relations A 2 = Band AB = E hold under multiplication; the group table may be written as
E
A
B
E
E
A
B
A
A
B
E
B
B
E
A
b) A cyclic group of order 3 comprises the members Z, Z2 (= Z-1) and Z3 (= E) under multiplication; the group table becomes
E
z
E
E
z
z
z
E E
z
c) Finally, consider the group of order 3 comprising positive integers modulo 3, with addition as the law of combination; the group table is as follows:
a
1
2
a
a
1
2
1
1
2
2
2
a
a 1
These three groups are isomorphic. There is a one-to-one correspondence between the members of the groups: A ~Z B ~Z2 C ~Z3
~I
~2 ~O
The groups are also Abelian; this nature is revealed through the symmetry across the principal diagonals of the bodies of the group multiplication tables. We note also that no member of a group is repeated among any row or column within a group table.
42
Group Theory and Point Groups
[eh.3
An important group in studying crystal symmetry is the translation group: the basic translations a, band c that define a unit cell in a three-dimensional lattice comprise an infinite group of the form (3.7)
The law of combination is vector addition, and the zero vector (nl = n: = n3 = 0) represents the identity operation; the negative signs on n, introduce the inverse members of the group. The power of the group multiplication table enables us to write out the tables without necessarily knowing first all the relations between the members. For example, in Problem 1.2(d) we found a C2 axis and two perpendicular mirror planes intersecting in that axis. We denote these planes by cry and o ~, and we can immediately set down a partial group table of the corresponding symmetry operations: a, cr, E C2 E
E
C2
C2
o; a,
C2
cr,
cr,
cr, crv
It is not difficult to enhance it immediately, because of the nature of C2 and o themselves; each operation is its own inverse: E C2 cry crv '
c, or C 2cr, = a v ' . If we choose the former then column 3 within the table would contain o, twice, which is not permitted; hence, we must write C2 crv = o, , whereupon the remainder of the table can be completed: C2v E av av C2 We now have to decide whether C2cr,
=
E
E
C2
cr,
av
C2
C2
E
av
cry
o,
av
crv
E
C2
cr v
av
o;
C2
E
We shall refer to this table again; it relates to the point group C2v (C, 'cyclic'), which is Abelian, order 4. It is of frequent occurrence: for example, water, tetrafluorosulfur, chlorobenzene and the cis-dibromodichloroplatinum (II) anion all share this group.
Sec. 3.2]
What is grout) theory
43
3.2.4 Subgroups and eosets We introduced subgroups briefly under Section 3.2.2, and here we discuss them more fully. The group multiplication table for ~v above indicates three subgroups: the identity group £.{E}, or C1 {E}, the group C2 {E, C 2} and the group C. {E, o}, If we remove all reference to the operation C 2 in ~v, that is, C 2 and one c, because C 2 = o,o ~ , we are left with the operations E and cr. They form the subgroup C. :
C. E
rr
E E
o
c E
There is only one way in which C. may be derived from C2v, for which reason it is called an invariant subgroup of its supergroup C2v . The other subgroups of C2v are also invariant, as shown by Figure 3.1. It does not follow that these subgroups are necessarily invariant with respect to other supergroups, Figure 3.1 (see also Problem 3.4). The order g of a subgroup is always an integral submultiple of the order h of the corresponding supergroup, that is, g = hln, where n is an integer greater than unity. The point group D2h may be written D2h{E , C 2(z), C 2(v), C 2(x ), i, cr(xy), cr(zx), cr(vz)}. One subgroup of D2h is D2 {E, C2 (z), C 2 (v), C 2(x ) } . We can form the product of each member of D2 with a member of D2h that is not also a member of D2, say, cr(xy). If we carry out the right-multiplication of D2 by the set D2cr(xy ) obtained is termed the a right coset of D2 : D2 cr(xy ) = Ecr(xy), C 2 (z)cr(xy ) , C 2 (v)cr(xy), C 2 (x )cr(xy ) = cr(xy),
i, cr(vz), cr(zx).
Thus, the group D2h can be represented by the combination of D2 and the right coset of its subgroup D2 with cr(~v) under addition:
D2h = D2 + D2cr(~V). Left cosets may be constructed in an analogous manner.
44
Group Theory and Point Groups
A~
v
AI' ." 0
[Ch.3
"
~
1
Fig. 3.1 The thirty-two crystallographic point groups and their subgroups; thin lines indicate the subgroups that are invariant. Thus, C2v is an invariant subgroup of D2d, C4v and D2h, but not of C6v or D3h
3.2.5 Symmetry classes and conjugates Subgroups provide one method for separating the members of a group into smaller sets, each constituting a group. An alternative procedure introduces the topic of symmetry classes. In Section 2.5.4, we introduced the similarity transformation; we now use this concept to discuss symmetry classes. If two symmetry operations A and B of a group are linked to a third operation R by the equation B = R1AR,
(3.8)
then B is the similarity transformation of A by R, and A and B are said to be conjugate to each other. Conjugate members of a group have useful properties, as follow: a) Every member of a group is self-conjugate, that is, for any member A A = R1AR:
(3.9)
for left-multiplication with A'I leads to A,IA = E = A,IR1AR = (RAYI(AR),
(3.10)
following Section 2.5.3. It follows that RA = AR so that R and A commute; R may be just E, which is always present, or any other member that commutes with A.
Sec. 3.2]
45
What is grout) theory
b) If A is conjugate to Band B is conjugate to C, then A is conjugate to C. We have
A
R'BR,
(3.11)
B = Q-'CQ.
(3.12)
=
Substituting for B in (3. 11) leads to A = RIQ-1CQR = (QRrlC(QR) = Z-ICZ,
(3.13)
where Z (= QR) is also a member of the group, conjugate to A. c) As a corollary to (b), by putting C = A in (3.11H3.12), it follows that if A is conjugate to B, then B is conjugate to A. We define a symmetry class to comprise those members of a group that are conjugate one to the other. This definition may be illustrated with the aid of the group tables that we have already discussed. We take the cyclic group of order 3, C3{E, Z, Z2}, and perform a similarity transformation of each member by each member of the group, with the following results: E: K1EE = EE = E Z-'EZ = Z-I Z = E (Z2r'EZ2 = (Z2r 1Z2 = E Z: K1ZE = ZE = Z Z-'ZZ = EZ = Z (Z2r IZZ2 = (Z 2rIE = Z Z2: K 1(Z2)E = Z2 Z-I (Z2)Z = Z-I E = Z2 (Z 2r' (Z2)Z2 = EZ2 = Z2 It is evident that this cyclic group contains three symmetry classes, each with one member, E, Z or Z2. It is left as an exercise to the reader to show that the group ~y {E, C2, o, , 0" ~ } contains correctly the four symmetry classes E, C2 O"y and 0" ~ .
We shall content ourselves with one more example group, that exhibited by the trigonal-pyramidal ammonia molecule and symbolized by C3y; its order is 6 and the group multiplication table is given below. We work through one member, C 3, and leave the remainder as an exercise for the reader (see also Problem 3.5). C3y
E
C3
C~
O"y
O"y
o"v
E
E
C3
C~
O"V
O"y
O"y"
C3
C3
C~
E
O"y
O"y"
O"v
2
C~
C :J
E
C3
O"y
O"V
O"y
0",.
"
O"V
O"y
O"y
E
C~
C3
O"y
O"y
0",.
O"y"
C3
E
C~
"
O"y"
O"y
O"y
C~
C3
E
O"y K IC3E = C3 (C3r C 3C3 = EC 3 = C 3 l
(C ~
r'c, C ~
= (C~
rlE
C3
"
O"v-l C 30"v 0" v'I- C30" 'v
O"v_IO" y = O"vO" v =C~
O"~ -IC30"~
0" ~ O"V = C ~
O"yO"y" = C~
46
GrOUI) Theory and Point GroUI)S
[eh.3
From the results just obtained, it follows that the operations C 3 and C:1 belong to one and the same class in this group. In a similar manner, cry, o v and o ~ form a single class, and we write the classes for C3y as E
The orders of classes must be integral submultiples of the order h of the group. In group C3 , the classes are E, C 3 and C 3
:
the operations C 3 and C:1 do not combine
into one class in this group; there is no other operator in the group that will combine with C 3 to give C ~ . In fact, all symmetry classes of Abelian groups comprise single symmetry operators. We need not necessarily engage in the labour of performing these similarity transformations each time we need to determine the symmetry classes of a given group. It will be sufficient to separate those operations that are themselves equivalent under a symmetry operation of the group. Thus, in the group that we have just considered, the three cry operations are equivalent under C 3 , whereas C 3 and C 3 are equivalent under the cry operations: E itself always forms a separate class in all groups; 0- 1EO = E for any chosen operator O.
3.3 DEFINING, DERIVING AND RECOGNIZING POINT GROUPS We have remarked already that Figure 1.4 is an example illustration of group closure, Section 3.2. I: two symmetry operations in a set were combined to produce another operation, also in the set. A collection of symmetry operations for a body is referred to as a point group, which may be defined formally as a set ofsymmetry operations, the action of which leaves at least one point of the body invariant, or unmoved. A similar definition may be expressed in terms of symmetry elements. All symmetry elements of a point group pass through the invariant point. In some cases it is a line or even a plane that remains invariant under the symmetry operations of a point group. The origin of the reference axes must be a point of invariance under all symmetry operations: otherwise, there would arise a multiplicity of origins (and symmetry elements), which is a feature of repeating patterns. Figure 3.2 illustrates the effect of two non-intersecting (parallel) twofold axes; it is clear that they lead to infinite repetition, to patterns that are governed by space-group theory. 3.3.1 Deriving point gmulls The operation of rotation C, (Section 2.2.3) is defined formally for values of n from I to infinity. However, to keep the discussion to a comfortable length, we shall restrict ourselves at first to the thirty-two crystallographic point groups, which means that n can take the values L 2, 3, 4, and 6. Subsequently, we shall consider some extensions ofn, one of which is explicit in Figure 2.2.
Sec. 3.3]
Defining, deriving and recognizing point groups
47
- - - .... 00
+ 1
+
+
00
00
00
2
3
a
5
4
c
b
6
e
d
Fig. 3.2 Effect of two parallel twofold axes (diads) a and b, considered to be lying in the plane of the diagram. Point 2 is related to point I by rotation about axis a; the ± signs indicate a given, fixed distance above or below the plane of the diagram. Points I and 2 rotated about axis b produce points 3 and 4; but 3 and 4 are, themselves, related by another diad, c. The effect of diad c on points I and 2 is to produce points 5 and 6; but they are related to 3 and 4 by diad d, and to each other by diad e. Now 3 and 4 can be rotated about e, and so on. This progression would lead to an infinite number of parallel, equidistant diads, together with the symmetry-related points, an arrangement that is wholly inadmissible under point-group synunetry.
The simplest point group consists of the single symmetry element C\, which corresponds to the identity operation E, or C\: we give this group the symbol C\, and its order h is I. Evidently, we can write another four such simple groups Cn with n = 2,3,4 and 6; we will refer to these groups as type (a). Another five groups are S, (n = 1, 2, 3,4 and 6). We note here that it is conventional to refer to 5\ as C. (it has a single symmetry plane), to 5:l as C, (it has a point of inversion, or centrosymmetry), and to 56 as C3h (it has a C3 axis normal to a symmetry plane); we call these groups type (b). In order to proceed further. it is necessary to consider combinations of symmetry operations: it is evident that combinations such as EC n and ES n cannot lead to any new groups, because E is a member of all groups, so we consider next the combinations iC n .. For n = 1. iC\ = i, which was derived under type (b), as point group Ci (5:l). For n = 2, we have i combined with C 2 . We define a group 6{E, C 2 , i, ...}, and we can construct a partial group table:
E C2
E
C2
E C2
C2 E
i E
It is not difficult to see that the combined effect of rotation by 180 0 and inversion through the centre of symmetry. a point on the C2 axis, is equivalent to reflection, from the initial position, through a symmetry plane normal to the C2 axis. Because of its orientation this plane is designated Gh. The point-group symbol is C2h, and we can now complete its group table:
48
Group Theory and Point Groups
C2h
E
E
E
C2
C2
C2 C2
ah ah
E
all
E
all
ah
ah
[Ch.3
C2
i
i
C2
E
We can show the effect of the combination iC 2 graphically by means of a stereogram (see Appendix 3). In Figure 3.3, the points have been enumerated, and we can trace the following operations:
i ~3
2
~
4
We see that the operation C2 followed by i is equivalent to the reflection operation we write this result as
ah;
(3.14) In this example, the combination C 2i gives an equivalent result; i and C 2 commute. We recognize this equation as another example of closure (Section 3.2.1). This group may be written as C2h {E, C 2, i, all}; h = 4, and its subgroups are C\, C2 , C, and C, . We stress that we symbolize the reflection plane in this example as ah even though it is set vertically with respect to the stereogram: the symmetry plane normal to the principal C, axis is always designated ah, whatever the orientation of the stereogram or other illustration of the group. Conventionally, the z axis (C 2 ) would be set normal to the plane of the stereogram: the variation adopted here allows the four symmetry-related points to be appreciated more readily on the diagram. Proceeding in this manner for other values of Cn, we find that i combined with C, generates all if n is even. When n is 3, i combined with C 3 generates 8 6; the result
Fig. 3.3 Stereogram lor point group C2h; the C2 axis (z) lies lett to right on the diagram, and the O'h plane (x,y), indicated by the heavy line, is normal to C2.
Sec. 3.3]
Defining, deriving and recognizing point groups
49
for n = I has been discussed already. Thus, the combinations of C, and i have led to three new point groups, C:l1" C4h and C6h , denoted as type (c). Next we may consider the combinations is,,. lt is not difficult to show that no point groups are obtained that are not already included under types (b) and (c), and we summarize the thirteen point groups derived so far in Table 3.1. Note the conventional symbols for certain S, groups in type (b); the symbols in parentheses indicate second occurrences of a group within the table. Table 3.1 Type
a) b) c)
Partial set from the 32 crystallographic point groups
Operator/s
Cn Sn
ic,
Number of groups
5 5 3
Point-group symbols n 4 3 2
C1 C, (C i )
C2 C, C2h
C3 C3h (56)
C4 54
C4h
6
C6 56
C6h
Euler's construction In order to derive further point groups, we must combine C, operations with operations other than E or i. Until now, there has been no problem about the relative orientations of the symmetry elements that we have used; i has been a point on an axis. However, if we wish to combine, say, C, and C 2 , we need to know their relative orientations. It seems likely that a C2 axis would lie perpendicular to Cn, or even collinear with it: but are these orientations correct, and are there other possibilites that we should consider? We shall discover that constraints exist that limit the number of possible combinations of symmetry elements. whether or no we are restricting the discussion to just the crystallographic point groups. It is instructive to carry out the necessary analysis by a construction due to Euler. In principle, it is another application of the group law of closure. of which (3. 14) is one example, as we have seen. We may consider Euler's proposition first in terms of proper rotations Cn, and then extend it to include the improper rotations Sn. A geometric illustration of Euler's analysis is illuminating. Let OA and OB (Figure 3.4) represent two symmetry axes intersecting at 0, the centre of a spherical projection of a symmetrical crystal (see Appendix 3); A, Band A' define the equatorial plane, normal to the direction through C, 0 and C'. Let BA C = BA C' = ai2, and ABC = A iJ c = ~i/2, where a. and ~ are angular rotations associated with the axes OA and OB respectively. Consider the motion of the line Ot'. The anticlockwise rotation «, about OA, maps C on to C; the anticlockwise rotation ~1, about OB, returns C to its original position. The sense of rotation given here is for observation along the axis towards the origin O. The arcs CBC and CAe' are zone circles (q.v.)for the axes OA and OB respectively. We can think of C' as the image of C in the plane OAB. lt may help to consider Figure A3.2 (Appendix 3), where OP would map on to OP' by an anticlockwise rotation of 1800 about the axis bOb'; a 90 0 rotation would have taken
50
Group Theory and Point Groups
[Ch.3
P just to the point e in the equatorial plane. Similarly, an anticlockwise rotation of 180° about eOe' returns P to its first location.
A'
Fig 3.4 Partial spherical projection of symmetry directions in a body; OA and OB are the given rotation axes.
The combination of the two rotations, 0.(04) and ~(B), leaves the point C unmoved: consequently, if there is a motion of a point on the sphere arising from the combination of the given two rotations, then the third, resultant symmetry element must pass through C, 0 and C. Consider next the motion of the point A under the same two operations, The rotation 0.(04) leaves point A unmoved: the rotation ~~(B) maps A on to A', where ABC = A' B C = ~/2; A' is, thus, the image of A in the plane 0 Be. In the spherical triangles ABC and A'BC, ABC = A' B C = ~V2, AB = A'B, and BC is common to both triangles. The triangles are congruent, so that A CB = A' CB. Let these angles be y/2; then the anti clockwise rotation y, about OC, maps A, along arc ABA', on to A', Again, we consider Figure A3.2: under rotation o.(A), about bOb', point b is invariant, whereas the rotation ~(B) about eOe' moves b to b', We may write the result symbolically as ~(B)
o.(A) = y(C),
(3,15)
which means that a rotation ex. in one sense about a symmetry axis OA followed by a rotation ~ in the same sense about an intersecting symmetry axis OB is equivalent to a rotation y in the same sense about a mutually intesecting symmetry axis OC (or clockwise about OC), where OA, OB and OC form a right-handed set of axes. We can use triangle ABC to solve for the angles a, band c between the three pairs of symmetry axes OB OC, OC 004, and OA OB; it is the reflection of triangle ABC in the plane AOB. The solution follows the equations for the polar triangle [Appendix 3, (A3.11) and its cyclic permutations], and the relevant portion of Figure 3.4 is shown in Figure 3.5. We know that a (= 24), P (= 2 B) and y (= 2 C) can take only the values 360° (0°), 180°, 120°,90° and 60°, corresponding to n = I, 2,3,4 and 6 in Cn.
Sec. 3.3]
Defining, deriving and recognizing point groups
51
(ii) The value of 360° for 0., p and y is ignored, because it corresponds to the trivial onefold rotation, or identity; (iii) Since we need only the number of combinations of symmetry elements, permutations are ignored so that only those solutions for 0. = 180°, 120°,90° and 60°, with Ps 0. and y S; p, are required;
(iv) Solutions for which one or two of a, band c are zero are ignored, because they correspond to a dimensionality in the problem of less than three.
B
A
P/2
S/2
c
Fig. 3.5 Spherical triangle ABC, to be solved for a, band c
Subject to these conditions, the six nontrivial results listed in Table 3.2 are derived. In interpreting this table, we recall from Euler's construction that 0., ~ and y refer, in order, to the anticlockwise rotations about the symmetry axes OA, OB and OC', with a, b and c being the angles between the pairs of axes OB OC', OC' OA and OA OB, respectively. Thus, in type (e), for example, we have two C2 axes (0., P) with a C4 axis (y) normal to them, the angle c between the C2 axes (OA and OB) being 45°. Figure 3.6 illustrates the relative orientations of the symmetry axes, and the stereograms show the effect of the symmetry operations on a typical point or vector in an object ofthe corresponding symmetry. In the stereograms the vertical direction (the normal to the stereogram) is the z axis and, except for Figure 3.6(h), corresponds to the rotation axis of highest degree in the group. The y axis is the left to right direction, and the x axis is perpendicular to both y and z. In Figure 3.6(h), the x, y and z axes are C2 symmetry axes, with C3 axes along the body-diagonal directions of a cube; in Figure 3.6(i), the x, y and z axes are each C4 symmetry axes with C3 axes again along the cube diagonals. Where are the C2 axes in this example? In those groups for which there is a single axis of threefold or sixfold symmetry, the orthogonal reference axes do not all coincide with the symmetry axes. However, in studying crystal symmetry, it is conventional that the crystallographic reference axes are taken along the directions of the symmetry axes, and we shall consider this situation more fully in Chapter 8.
52
[eh.3
GrOU)l Theory and Point Groups
Table 3.2 Results following from Euler's construction Type
«/deg
Wdeg
yldeg
aldeg
bldeg
c1deg
d)
180
180
180
90
90
90
e)
180
180
90
90
90
45
f)
180
180
120
90
90
60
g)
180
180
60
90
90
30
h)
120
120
180
cos" (11...)3)
cos" (1/...)3)
i)
90
180
120
cos"(...)6/3)
cos" (1/...)3)
cos" (1/3) 45
3.3.2 Building up the point groups Now that we have determined the angles at which C, symmetry axes may intersect, we can list the point groups corresponding to the types given in Table 3.2. For types (d) to (i) they are, in order D2 , D4 , D3, D6 (D, dihedral), T (tetrahedral), 0 (octahedral). We explore now the combinations of C2 and S!, but before we commence this analysis we note the following general combinations, which may be deduced readily either by application of (3.1) under multiplication, or with stereograms: (Proper rotation) (Proper rotation) = Proper rotation (Proper rotation) (Improper rotation) Improper rotation (Improper rotation) (Improper rotation) = Proper rotation
=
For type (d) in Table 3.2, the combinations of mutually perpendicular Cn and 8 1 axes listed in Table 3.3 are permitted. The first entry in Table 3.3 lists three mutually perpendicular twofold axes, and the point-group symbol for this combination is D2 , which we have already noted (Figure 3.6). Next we replace two (Why not just one?) C 2 axes by S\ axes. We have seen that 8 1 is equivalent to a a reflection plane normal to the direction of the SJ axis, and we have actually considered this combination already in Section 3.2.3; it corresponds to point group C2v. No other combinations are possible for type (d). However, we must always consider if we have formed a group in each type that contains a centre of symmetry, that is, it includes the operator i. Neither D2 nor C2v is centrosymmetric, so now we combine the operation i with D2 , giving the partial group D2h {E, C 2 (z), C 2 (y), C 2 (x), i, ...}. We know already that iC 2 is equivalent to a a plane normal to C2 : thus, we have further the operators a(xy), a(zx) and a(vz) perpendicular to C 2 (z), C 2 (y) and C 2(x), respectively. No new operators are produced by any other combinations within this group: thus we obtain the character table below. We see that this point group is Abelian (all pairs of symmetry operators commute) and its order is 8. Its symbol, D2h, shows that it can be obtained from D:! by combining a reflection plane normal to a C2 axis in D:!. This result may be confirmed by comparing the stereograms for D2 and D2h in Figure 3.7. The combination of i (or ah) with the
Sec. 3.3]
Defining, deriving and recognizing point groups
53
45°
+0
-0
d
e
f
h
Fig. 3.6 The permitted angles between rotation axes C;
9
54
[Ch.3
GroUI) theory and point groups
Table 3.3 Permitted combinations of C2 and SI for type (d) OA (x) axis
OB (y) axis
C2
OC' (z) axis
C2
s,
C2 C2
s,
D2h
E
C 2 (z)
C 2 (y)
C 2 (x )
E C2 (z)
E C2 (z)
C 2 (z) E C 2(x )
C 2(x>: C 2 (y): C2 (y):
C2 (x)
C2 (x)
C 2 (y) C2 (x) E C2 (z)
cr(vz)
o'(zx) cr(xy)
C2 (y)
cr(xy) cr(zx) cr(yz)
C2 (y)
................ cr(xy) cr(zx) cr(yz)
C2 (y) ~(~yj
cr(vz) cr(zx)
E
.. .'cr(zx) ""(j-(Vi)' cr(xy)
Point-group symbol
cr(xy) cr(zx) o (vz)
E C2 (z)
C2 (y)
C2 (x)
D2
C2v
cr(xy)
cr(zx)
cr(yz)
cr(xy)
cr(zx) cr(Yz)
cr(Yz) cr(zx) cr(xy)
o'(pz)
cr(zx) C2 (z) E C2 (x)
C2(y)
cr(xy)
C2 (y)
C2 (x)
C2 (x) E C2 (z)
C2 (z) E
C2 (y)
symmetry elements of D2 introduces all remaining symmetry elements of D2h . If we repeat this exercise with C2v• we find that its combination with i (or crh) again produces D2h . These results may be summarized by the scheme +i or rr.. ~
Direct products of groups We may express these relationships neatly in terms of a direct product of groups, either D2 or C2v with Cj . The group multiplication table for D2h (above) may be
m \±] (a)
(b)
Fig. 3.7 Stereograms tor point groups (a) D2, and (b) D2h: the introduction of the centre of symmetry into D21eads to mirror planes perpendicular to all C2 axes ofD2.
divided into quadrants, that outlined in the table being the group multiplication table for D2 . A direct product of D2 and Cj is obtained by forming all possible products between the symmetry operations of the two groups, which we may write as D2 (8) C, = D2h or, more conveniently
(3.16)
Sec. 3.3]
Defining Point Groups
55
where the direct product symbol 0 implies all possible products under multiplication between the symmetry operations of O2 and i. It is not difficult to see that the following direct products also lead to point group 02h: O2 0 ah = C2V 0 i = C2v 0 ah =
02h
(3.17)
02h 02h
C 2 in the third equation (3.17) acts in a direction normal to ah (see also Section 3.2.4). We consider one more example of point-group derivation from the data in Table 3.2, namely type (f), The axes OB and OC' correspond to the reference axes y and z respectively, whereas the x axis lies between two C2 axes, as shown in Figure 3.8a. Although only three symmetry axes were needed in Euler's construction, a C2 axis normal to a Cn axis leads to a total of n C2 axes that are equivalent under the symmetry C; Table 3.4 indicates the permitted combinations of one C3 axis with C2 and SI axes. In point group 0 3 , there are three C2 axes in the xy plane, normal to the C3 axis (z) and related by it. The group may be written formally as 0 3 {E, C3 , C:i , C 2, C 2" Cs-}; its order is six, and the group multiplication table is easily constructed. The stereogram in Figure 3.8a illustrates this point group; it appears also among the six unique solutions in Figure 3.6. If we replace the C2 axes by SI axes then, because the Sl axes are normal to the C3 axes, the symmetry planes to which they are equivalent are vertical (3a v) and intersect in the C3 axis. Thus, we have the group C3v {E, C 3 • C:3 , a v , a v , a ~ }:
its order is 6, and its group table has been given in Section 3.2.5. Each of the point groups O2 and C3v combines with i to form D3d : +i (-
The stereograms in Figure 3.8 illustrate these point groups and their interrelationships. In this way we may continue to derive the remainder of the thirty-two crystallographic point groups: we would find that no roto-reflection axis other than SI is needed in this derivation; the other permitted S; axes emerge during the Table 3.4 Point groups derived from type (f) OA axis
OB (v) axis
OC' (z) axis
Point -group symbol
Group theory and point groups
56
[eh.3
--> OY • Y axis
1
X
ax t e
(b) Fig. 3.8 Stereograms to show point groups (a) D3, (b) C3vand D3d; D3d may be formed from D3 or C3v by direct product with Cj ••
derivation. A summary of the crystallographic point groups, together with the two groups for cylindrical symmetry (linear groups), is provided in Table 3.5. A program, EULS (EULH in the Hermann-Mauguin notation), is available on the Internet Web site www.horwood.net/publish (from where further user directions are obtainable) that leads the user through the various stages of Euler's construction as discussed above; it considers also certain noncrystallographic point-group symmetries. The reader is encouraged to make use of this facility. Table 3.5 Point groups and their symmetry operations Group
Operations
C1
E
Cz
E,
C3
E, C3"'C~
C4
E, C 4
CZ
Cs
E,Cs
C;
C6
E, C 6
C 3 ••• C Z... C ~ ... C ~
Cn
:
One n-fold (proper) rotation axis
c, C~ C~ ...
ct
C.: One mirror symmetry plane (a)
c,
E, a Cj
:
One point of inversion (centrosymmetry)
Sec. 3.3]
c.
Defining Point Groups
57
E, i S, : One n-fold roto-reflection axis (n even)
54
E, S4 .. ,S2."S ~
56
E, C 3 ... C i
... 8 6 ... 8 ~
,i
Cnv : One C, axis and n vertical c, planes intersecting in C n
C3v
E, C2 , 20 v E, C3 C i , 30v
C4v
E, C 4
Csv
E, 2C s
C6v
E, C 6 C 3... C 2 ... C ~ , C ~ , 30v , 30d
C2v
C 2 ... C ~ , 20v , 20 d
2C; , 50v Cnh : One C, axis and one 0h plane normal to C«
C3h
E, C 2 , i, 0h E, C3 C~
C4h
E, C 4
C2
CSh
E, C,
C;
C6h
E, C 6 C 3 C 2 ... C 3
~h
ah
83"'S~, C~
8 4 ... 5 ~ . i, 0h
C ~ ... S5
C~
... C 6
5 ~ ... 8; 8 6 ... 8 3 S 3
8~. , 0h ... 8 6 ,
l, 0h
On: One C, axis and n C2 axes normal to it
E,3C 2 E, C3"'C~,
E, C 4
3C~ C 2 ... C ~ , 2C ~ , 2C ~
E, 2C 5 2C ~ ... 5C ~ 5 E, C 6... C 3... C 2 ... C "'3 ... C 6' 3C •2, 3C "2
0nh: As On ' plus one (horizontal) 0h plane normal to C;
E, C 2 (z), C 2 (y ), C 2 (x ), i, o(xy), o(zx), o(yz) E, C 3 C ~ E, C 4
C2
S3 S ~ , 3C C~
'2'
ah, 30 v
S4... S ~ , 2C ~ , 2C ~ , i, 0h, 20 v , 20 d
E, 2C s 2C ~ ... 2S ... 2S ~ ... 5S ~ , 0h, 50v E, C 6... C 3... C 2 ... C 30d
2 5 "S'" • 3 ... C 6 ... 5 6 ... 5 3...5'3 ... 5 6 , 3C 2' 3C 2' I,
0h, 30v ,
Ond : As On, plus n vertical (dihedral) ad planes intersecting in Cn
Group theory and point groups
58
1
[eh.3
'
E, C Z••• S4 S 4, 2C 2' 2ad
E, C 3 .•. ci
,3C'2 .i, 3ad
S6"'S~
E, 2S8oo .2C 4°o .s.. .2S R , 4C ~ ,4ad
Cubic groups: T, tetrahedral; 0, octahedral. T
r, r, o 0h
E, 4(C 3 • ooC i), 3C z
i ",S6°o' S ~), 3C z. i, 3ah E, 4(C C i), 3(C ooS4°o'S 1)6 a d E, 4(C 3°o'C D, 3(C 4oo.C zoo. C 1). 6C 2 E, 4(C 3°o'C i oo,S6°o'S 6), 3(C C 1.·S4oo ,S 1). 6C 'z, i, 3ah, 6ad E, 4(C 3.ooC 3°o'
2.
4oo.CZoo.
Coov :
One C" axis and an infinity of a v planes intersecting in
Dooh : As C oov, plus one ah, plane normal to Dooh
E, (C~
.ooC ~vl ... S ~ ... S ~vl),
Coo
Coo
ooC ~ ,i, ooav
Notes on Table 3.5
1) Symmetry operators linked by the dotted line ... act collinearly. 2) A vertical symmetry plane is designated ad if it bisects the angle between twofold axes that lie normal to the principal C n axis. In point groups C4v and C6v , the labelling of the ad planes is not strictly consistent with this rule, but the notation is general [lZ l . 3) In point group D4 , the symmetry element C2 ' lies along the x axis and C2 " between x and y; in D6 , C ~ is along y. 4) In point groups D4h and D6" . the o, planes contain the symmetry elements 2C ~ ,
5) 6) 7) 8) 9)
and ad the elements 2C'~ . Strictly. all planes are ad type but the conventional notation is used here. In the linear point groups Coov and D",h, ~ represents an arbitrary angle of rotation. The order of symmetry operations listed here will not necessarily be the same as that in the corresponding character tables. The point-group symbols C. and C, are preferred over S, and Sz. Of the five cubic point groups, Td and 0" are of particular significance in chemistry. The unprimed symbol C z refers to an operation along the z axis, or the x, y and z axes in the cubic point groups.
Sec. 3.3]
Defining Point Groups
59
10) The symbols C 2, and C 2" refer to different forms (q.v.) of C 2 operations, n of each, that are symmetry-related under the principal Cn operation. The symbols C 2, C 2" C 2 " are used to indicate the n C 2 axes in one and the same form. II) The parentheses in the cubic groups include operators with a common multiplying factor, but which are not necessarily of the same symmetry class. Thus, in r, 4(C 3 •.. C ~ ) leads to the class 8C 3 , whereas in Oh 4(C 3°o'C ~ .S6... S ~ ) leads to the classes 8C 3 and 8S6. 00
3.3.3 Federov and Plato solids Federov showed[62j that there are only five polyhedra, each type of which can be packed in one and the same orientation to fill space completely. Figure 3.9 illustrates the Federov solids; they are also Wigner-Seitz cells (q.v.), derived from Bravais lattice unit cells: Federov solid
Crystal form/s
Lattice unit cell
Cube Rhombic dodecahedron Cube + octahedron Hexagonal prism Elongated rhombic dodecahedron
{100}
Cubic? Cubic I CubicF Hexagonal Rhex Tetragonal I
{llD}
{100} + {Ill} {l0 1 O} {l00} + {lII}
If the program EULS (or EULR) has been used in studying Euler's construction, it would have been noticed that, although the program extends the discussion given here to include the noncrystallographic rotation axes Cs, C7 and Cg, no point groups emerged for which there were intersecting rotation axes of degree greater than 5, as in point group I, for example. This result is related to the existence of only five regular solids, the Plato solids, that is, those with regular polygonal faces, and among which the maximum proper rotation axis found is Cs: a regular polygon has equal sides and equal angles between successive sides.
(a)
(b)
(c)
(d)
(e)
Fig. 3.9 Federov solids: (a) cube, (b) rhombic dodecahedron, (c) cube + octahedron, (d) hexagonal prism, and (e) elongated rhombic dodecahedron. The cube is also a regular, Plato solid.
The Plato solids may be determined from the following argument. Consider a regular polygon with p sides, and let q such polygons form the comer of a polyhedron. Since the internal angle of a poly-p-gon is [90(2p - 4)/p]0 and the sum of the angles formed by the faces at any comer must be less that 360°, it follows that q(2p - 4)/p < 360/90,
(3.18)
GrOUI) theory anti point groups
60
[eh.3
which simplifies to
l/q + lip> \12.
(3.19)
Euler showedl13] that the numbers of faces f, edges e and corners c of a polyhedron are related by the equation c + f= e + 2.
(3.20)
Since, p and q must each be greater than 2, the only values of p and q that satisfy inequality (3.19) are those listed in Table 3.6. The less familiar pentagonal dodecahedron and the icosahedron are shown in Figure 3.10, and both exhibit Cs and S\o symmetry axes.
(a)
(b)
Fig. 3.10 (a) Pentagonal dodecahedron, (b) Icosahedron
Table 3.6 Values for p and q in (3.19), and the Plato solids p,q 3,3 3,4 4,3 3,5
e
5,3
30
6 12 12 30
f
c
4 6 8 12
4 8 6 20
20
12
Plato solid Tetrahedron Cube Octahedron Pentagonal dodecahedron Icosahedron
Point group
Td
°h °hI h
Ih
These two solids are related to each other in the same way as are the cube and octahedron, that is, by an interchange of the numbers of faces f and corners c. The 3,5 and 5,3 solids have the same symmetry, and their analyses are summarized in Table 3.7. From the symmetry axis SIO there are the operations E, 510 , 5 ~o
c ~ , 5 io =
=
C s, 5 fo , 5
to =
= i, S fo = 5; , S io , S fa = C ~ and S io ; from Cs we obtain E and C 5 (n
1-4), all included with the SIO operations. From S6 we obtain the symmetry operations E, 56, 5 ~
= C3 ,
S ~ =i, S:
=C~
and S ~ : under C3 we have the operations E, C 3 and C ~ , which are included with the S6 operations. There are also fifteen C ~ operations and fifteen a operations in each solid.
Sec. 3.3]
Defining Point Groups
61
Table 3.7 Analysis of the pentagonal dodecahedron and icosahedron Symmetry element
Orientation of symmetry element Pentagonal dodecahedron Icosahedron
Six SIO through:
centres of opposite pairs of pentagonal faces
opposite vertices
Six Cs collinear: Ten S6 through:
with SIO opposite vertices
with S,
Ten C3 collinear: Fifteen C 2 bisecting:
with S6 pairs of opposite edges
with S6 pairs of opposite edges
Fifteen cr containing:
two C s and two C2
two C s and two C2
centres of opposite trianglar faces
If we analyse these data in the customary manner, we would deduce the following symmetry classes for the pentagonal dodecahedron and the icosahedron: E, 12C s, 12C; , 20C 3 , 15C ~ , i, 12S10 , l25?o, 205 6 , 15cr. The point group is known as Ih and its order is 120. If we excise all reference to the centre of symmetry, that is, all 8 10,56 , cr and i, we obtain the classes E. l2C s, 12C ~ , 20C 3 , 15C ~ , which relate to the pure rotation subgroup I. We shall not meet examples of point groups I and Ih in our study, but they are encountered with some molecules of biological interest. 3.3.4 Practical recognition of point groups In the scheme for practical point-group recognition to be described here, molecules and crystals are divided first into four groups depending upon the presence of a centre of symmetry, or a mirror plane, or a centre of symmetry and a mirror plane, or neither of these two symmetry elements, leading to Table 3.8. The recognition of a centre of symmetry or of a mirror plane is very straightforward: for a centre of symmetry, place the given model on a flat surface; then, if the plane through the uppermost atoms (in the case of a molecular model) or the uppermost face of the crystal (in the case of a crystal model) is parallel to the supporting surface, a centre of symmetry is present in the model. If a mirror plane is present, it divides the model, conceptually, into halves that are related as an object is to its mirror image, the well known right-hand-s-left-hand enantiomorphism. It is suggested that the reader examine these simple rules with models of SF6 (or a cube) and CH4 (or a tetrahedron), or other examples. A correct identification at this stage is important: it places the model into one of types I-IV, shown by Table 3.8 and Figure 3.11. Further study concentrates on the nature of the principal symmetry axis, the number of them, if more than one, the presence of other mirror planes or twofold axes and so on, according to the scheme illustrated in Figure 3.11. It will be apparent that this scheme resembles a flow diagram of a computer program'!". Indeed, the procedure can be carried out with the program SYMS which is accessible
62
Group Theory and Point Groups
[eh.3
Further study concentrates on the nature of the principal symmetry axis, the number of them, if more than one, the presence of other mirror planes or twofold axes and so on, according to the scheme illustrated in Figure 3.11. It will be apparent that this scheme resembles a flow diagram of a computer programl'"'. Indeed, the procedure can be carried out with the program SYMS which is accessible
tIl
~ ~ n
>
17
,o
(II)
"~
1.------+1
C
~
YLa'" .nd
'.'"
Yo.
l~
n
-
'="
9'O!~----101
I
4.7
n
No
I-I
-----+i T ,
~
)0
11
C_I
~
r ee
I ... I'
'I
I
i I
I I
I
ur n
' 1? -~--{~
~
56
II
~1r1~:o.Lcn?i~~
I
D
ae
I
! I I I
Fig.3.11 Block diagram of the point-group recognition scheme embodied in the program SYMS (and SYM the scheme under types (Il) and (IV), respectively.
64
GrOUI) theory and point groups
[eh.3
Table 3.8 Point-group recognition scheme: the basic divisions Type I Type II Type III Type IV C1 C2 c, C2v c, 56 C2h C4h C3
C4
C3v
~
~
~
~
~
~
T
0
Caw
D4
D6
D2d
C4v
C6h ~
~
~
~
~
r,
r,
o,
Dooh
under the WWW reference already quoted; a key to the link between a 'model number' and its point group is also provided. For best results the model should have been studied carefully along the lines indicated'" before using the program. If an error is made in the deduction process, the program returns the user to the point in the scheme at which the error has been made. Up to two such returns are allowed before the program offers a suggestion for alternative action. The thirty-two crystallographic point groups are illustrated in Figure 3.12, using the traditional stereogram notation. The two diagrams for each point group show the symmetry elements, and the general equivalent positions that are related by the point-group symmetry. General positions do not lie on any symmetry element: special equivalent positions exist on symmetry elements, and the number of them for a given point group is a submultiple of the number of general positions. For the point groups C2 , C, and C2h , the two diagrams given relate to the choice of the twofold axis: it may be taken along z, which is conventional in molecular symmetry, or along y, which is the universal crystallographic convention. Table 3.9 provides a comparison of point-group symbols in the Schonflies and Hermann-Mauguin notations, with example compounds for each group.
Defining, deriving and recognizing point groups
Sec 3.3]
65
0 CD8 ffiEB
00 ffiEB CD CD8 ffiEB i \ ffi ,:r, CDi CDS C1
(l)
C2
(2)
C4
(4)
(m(=2))
84
(4)
(2im)
C
-
c,
<
(1)
C
2h
1-1-1
C2
(2)
Cs
(m)
a •
"'-1/
a •
4h
(4Im)
• • a
a
_-¥-t VI~I ..........
CDCD CDEB ffi® -
D
C
2
2u
(222)
(mm2)
D4
(422)
C
(4mm)
4v
ffi~'
-
€B a a. • -
wEB C 2 h (21m)
D
CDri\ t-t-t \....1/
00 D 2 h (mmm)
o o
2d
00
\(IY
(42m)
0
0
I
~
11.. I I" ~.(~,--~ \
I~
00 .... ~ D 4 h (4Immm)
Fig. 3.12 Stereograms of the 32 crystallographic point groups: the 1st and 2nd settings for monoclinic crystals correspond to the unique, twofold axis along z and y, respectively. Both symmetry notations (Table 3.9) are used in this figure.
, I
I
i
66
Group theory and point groups
.® ·.-J-. \ j \,J I '\- ~
.. @ . ®@ o.
D3
0
(32)
.. •
•
••
-
•
•
:. .: ••
c6 u (6mml
[Ch.3
Defining, deriving and recognizing point groups
Sec. 3.3] Table 3.9
Synunetry notations and example compounds
Schonflies Hermann-Mauguin
Cj ~
C3 C4 C6 C, C. S4 S6 C2h C3h C4h C6h C2v C3v C4v C6v O2 03 04 06 02h 03h
Example or possible example
I 2 3 4
CHBrCIF, bromochlorofluoromethane H20 2, hydrogen peroxide H 3P04, phosphoric acid (CH3)4C4, tetramethylcyclobutadiene
6 -
C6(CH3) ~ , pentamethylcyclopentadienyl
I
m
-
C6HsCH2CH2C6Hs, dibenzyl C6H3Ch, 1,3,5-trichlorobenzene
4
[H 2P04L dihydrogen phosphate ion
3
[Ni(N0 2)6( , hexanitronickelate(II) ion CHCl=CHC I, trans-I,2-dichloroethene
-
21m 6
41m 61m mm2 3m 4mm 6mm 222 32 422 622 mmm 6m2 4
C 3H3N3(N3h, 1,3,5-triazidotriazine [Ni(CN)4f, tetracyanonickelate(II) ion C6(CH3k hexamethylbenzene C~sCI, chlorobenzene CHCh, trichloromethane [SbFsf , pentafluoroantimonate(V) ion C 6(CH2Cl)6, hexachloromethylbenzene C8H 14, cycloocta-I,5-diene [S206f, dithionate ion Co(H 20)4Cb, tetraaquodichlorocobalt C6(NH2)6, hexaminobenzene C6H4C12, 1,2-dichlorobenzene [C0 3f, carbonate ion
04h
-mm
[Aufira]", tetrabromoaurate(III) ion
06h
-mm
6
C6H6 , benzene
02d
42m 3m 23 m3 43m 432 m3m
ThBr4, thorium tetrabromide
03d T Th Td 0
Oh
In
m
C 6H12, chair cyclo-hexane C(CH 3)4, 2,2-dimethlypropane [CO(N0 2)6f , hexanitrocobaltate(III) ion CH 4, methane C8(CH3 ) 8, octamethlylcubane SF6, sulfur hexafluoride
Noncrystallographic point groups
67
68
Group theory and point groups D4d
82m
Cs
5
DSh
5m 10m2 5m
c; DSd
c., Droh
om
odm
[Ch.3
8 8 , sulfur CS(CH3 ) ; , pentamethylcyclopentadienyl CsHsNiNO, nitrosylcyclopentadienylnickel (CsHshRu, biscyclopentadienylruthenium (CsHshFe, biscyclopentadienyliron HCI, hydrogen chloride CO2 , carbon dioxide
PROBLEMS 3 3.1 Which pairs of operations C 2 , C 3, rr and E commute? 3.2 Construct a group multiplication table for (a) C2h and (b) D2d . Is either of these groups Abelian? 3.3 Construct a multiplication table for the cyclic group 6{E, A, B, C} of order 4, using the relations A = r, B = ?, C = r 3 . Which operation, other than E, is its own inverse? 3.4 Figure P3.1 is a stereoview of the dithionate ion, as seen along the s--s bond. (a) Determine the point group of this species. (b) Construct a group multiplication table for this point group. (c) Determine the subgoups of this point group. (d) Which, if any, ofthe subgroups is invariant to the point group of the species. 3.5 By means of similarity transformations, or otherwise, show that the three cr operations in C3v fall into one symmetry class.
Fig. P3.1 Stereoview of the dithionate ion, along the S-S bond.
3.6 From a consideration of group multiplications tables, show that the point group
54 cannot be obtained from any combination of symmetry operations, whereas 56is equivalent to a certain combination. Then, determine this combination. 3.7 What point groups are formed by the direct products of (a) D4 and i, (b) C4 and cr(xy), and (c) i and cr. 3.8 What point groups may be developed from type (e) in Table 3.2'1 3.9 Use stereograms to show that C4v combined with i leads to D4h . 3.10 Determine the point groups of each of the following chemical species from their stereoviews:
69
Problems
a
b
c
70
Group Theory and Point Groups
[eh.3
d
e
f Fig P3.2 Stereoviews of the structures of six chemical species: (a) l,3 ,5-triazidotriazine; (b) chlorobenzene (c) cyclohexane (chair); (d) methane; (e) octafluorotantalate(V) ion (square antiprism); (f) hexachloroplatinate(IV) ion.
Problems
71
3.11 Show that a group consisting of the unique members E, A, B, C, D and F cannot have any member more than once in any column or row within the table. 3.12 Show that if any two members P and Q of a group satisfy the relation P = then P and Q commute. 3.13 How many groups may be constructed, and what are their types, for (a) order 4, and (b) order 5? Draw up group multiplication tables and list any members that are their own inverses. 3.14 Which of the symmetry operations of the point group of the dithionate ion (problem 3.4) are conjugate one with the other? 3.15 Is it true or false that the following pairs of symmetry operations commute? (a) C,... Cn~ (b) C, and i; (c) cr(xy) and cr(vz),; (d) C2 (z) and C2(y); (e) C, and crh; (f) C 4 (z) and C 2 (x ); (g) S, (n even) and i. 3.16 Comment on the equation crb C 4 = 8 4 in its relation to 54, the point group for C~, or a regular tetrahedron. 3.17 Use the point-group symbol D4h to show the power of the Schonflies notation. Take the crh plane normal to the C4 axis and the C ~ axes in the c plane (along x and y), all indicated by the symbol, and show by means of a stereogram, or otherwise, that all other symmetry elements of the group are introduced. 3.18 Show that the members ma (m = -CXJ, ... -1, 0, 1, ...CXJ) form a group under a certain law of combination, and state that law. 3.19 This problem and the next are designed for use with the program SYMS (or SYMH). The numbers in parentheses are the model numbers for use with the program. Determine the point group for (a) a regular tetrahedron (23), (b) a tetragonal pyramid (87), and (c) a hexagonal prism (33). Give the results in both the Schonflies and the Hermann-Mauguin notations. 3.20 Determine the point group for the species (a) cyclohexane in the chair form (40), (b) 1,2,4-trifluorobenzene (71), and (c) trichlordeuteromethane (42). Give the results in both notations. 3.21 Draw four irregular but identical quadrilaterals on a thin card. Cut them out and arrange them to form a symmetrical figure. What is the point-group symmetry of the resultant figure? 3.22 Figure P3.3 can be used to construct a model of a rhombic dodecahedron, a Federov solid. Make an enlarged photocopy of the figure; a second sheet of A4 paper glued to the back of the photocopy adds a useful rigidity. Cut out the figure, lightly score the dotted hinge-lines and fold all portions in the same sense. With a quick-drying glue, attach the flaps to the faces, flap A to face A, and so on. What is the point group of the model? If you are using the program SYMS (or SYMH) for point-group recognition, this model can be numbered 10.
o'ro.
GroUI) theory and point groups
72
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[eh.3