M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 31
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A closer look at this table reveals that thecharacters of all irreducible representations are equal in C3 and C32 andalso in v , v , and v , respectively. Thus, according to rule 4 C3 andC32 form one class, and v , v , and v together form another class.A complete character table is given in Table 4-5 for the C3v pointgroup.
The classes of symmetry operations are listed in the upper row,together with the number of operations in each class. Thus, it is clearfrom looking at this character table that there are two operations in theclass of threefold rotations and three in the class of vertical reflections.The identity operation, E, always forms a class by itself, and the sameis true for the inversion operation, i (which is, however, not present inthe C3v point group).
The number of classes in C3v is 3; this is also thenumber of irreducible representations, satisfying rule 5 as well.Consider now the symbols used for the names of the irreduciblerepresentations. These are the so-called Mulliken symbols, and theirmeaning is described below, along with other Mulliken symbolscollected in Table 4-6.Letters A and B are used for one-dimensional irreducible representations, depending on whether they are symmetric or antisymmetricTable 4-4. A Preliminary Character Table for the C3v Point GroupC3 EC3C32vv⌫1⌫2⌫311211−111−11−101−101−101944 Helpful Mathematical ToolsTable 4-5. Complete Character Table for the C3v Point GroupC3E2C33A1A2E11211−11−10zRz(x, y) (Rx , Ry )x2 + y2 , z2(x2 − y2 , xy) (xz, yz)with respect to rotation around the principal axis of the point group.∗Antisymmetric behavior here means changing sign or direction.
Thecharacter for a symmetric representation is +1, and this is designatedby the letter A. An antisymmetric behavior is represented by the letterB and has −1 character. E is the symbol§ for two-dimensional, and T(sometimes F) the symbol for three-dimensional representations. Thesubscripts g and u indicate whether the representation is symmetric orantisymmetric with respect to inversion. The German gerade meanseven and ungerade means odd. The superscripts and are used forirreducible representations which are symmetric and antisymmetricwith respect to a horizontal mirror plane, respectively. The subscripts1 and 2 with A and B refer to symmetric (1) and antisymmetric (2)behavior with respect to either a C2 axis perpendicular to the principalTable 4-6. Symbols for Irreducible Representations of Finite GroupsDimension ofCharacter underSymbol(s)RepresentationaECnihC2 or 12311231−11–11−11–1a∗ABETAgAuAAA1A2BgBuBB B1B2EgEuC2 axis perpendicular to the principal axisAntisymmetry will be discussed in the next Section.Not to be confused with the symbol of the identity operation, which is also E.§TgTu4.5.
Character Tables and Properties of Irreducible Representations195axis or, in its absence, to a vertical mirror plane. The meaning ofsubscripts 1 and 2 with E and T is more complicated, and will notbe discussed here. The character tables of the infinite groups, C∞v andD∞h , use Greek rather than Latin letters: ⌺ stands for one-dimensionalrepresentations and ⌸, ⌬, ⌽ etc., for two-dimensional representations.It is always possible to find a behavior that remains unchangedunder any of the symmetry operations of the given point group.
Thus,there is always an irreducible representation which has only +1 characters. This is the totally symmetric irreducible representation, and itis always the first one in any character table.The character tables usually consist of four main areas (sometimesthree if the last two are merged), as is seen in Table 4-5 for the C3v andin Table 4-7 for the C2h point group. The first area contains the symbolof the group (in the upper left corner) and the Mulliken symbols referring to the dimensionality of the representations and their relationshipto various symmetry operations. The second area contains the classesof symmetry operations (in the upper row) and the characters of theirreducible representations of the group.The third and fourth areas of the character table contain somechemically important basis functions for the group.
The third areacontains six symbols: x, y, z, Rx , Ry , and Rz . The first three are theCartesian coordinates that we have already used before as bases fora representation of the C2h point group (see, p. 186). The symbolsRx , Ry , and Rz stand for rotations around the x, y, and z axes, respectively. A popular toy, the spinning top, is helpful in visualizing theconsequences of symmetry operations on rotation.
Let us work outthe characters for rotation around the z axis in the C3v point group(Figure 4-9a). Obviously, the identity operation leaves the rotatingspinning top unchanged (character 1). So does the rotation around thesame axis since the rotational symmetry axis is indistinguishable fromthe axis of rotation of the toy. The corresponding character is again 1.C2hEAgBgAuBu1111Table 4-7. The C2h Character TableC2ih1−11−111−1−11−1−11RzRx , Ryzx, yx2 , y2 , x2 , xyxz, yz1964 Helpful Mathematical ToolsFigure 4-9. (a) Applying the identity and the C3 operation to a rotating spinningtop; (b) Illustration of the effect of mirror planes on the rotating spinning top.Now place a mirror next to the rotating toy (Figure 4-9b).
Irrespective of the position of the mirror, the rotation of the mirror image willalways have the opposite direction with respect to the real rotation.Accordingly, the character will be –1.Thus, the characters of the rotation around the z axis in the C3v pointgroup will be:11−1Indeed, Rz belongs to the irreducible representation A2 in the C3vcharacter table. In other words, Rz transforms as A2 , or, it forms abasis for A2 .The fourth area of the character table contains all the squares andbinary products of the coordinates according to their behavior underthe symmetry operations. All the coordinates and their products listedin the third and fourth areas of the character table are important basis4.6.
Antisymmetry197functions. They have the same symmetry properties as the atomicorbitals under the same names; z corresponds to pz , x 2 − y 2 to dx 2 −y 2 ,and so on. We shall meet them again in the discussion of the propertiesof atomic orbitals.The term antisymmetry has occurred several times above, and it is awhole new idea in our discussion.
It is again a point where chemistryand other fields meet in a uniquely important symmetry concept.4.6. AntisymmetryAntisymmetry is the symmetry of opposites [13]. “Operations of antisymmetry transform objects possessing two possible values of a givenproperty from one value to the other” [14]. The simplest demonstration of an antisymmetry operation is by color change. Figure 4-10shows an identity operation and an antiidentity operation. Nothingchanges, of course, in the former whereas merely the black-and-whitecoloring reverses in the latter. Antimirror symmetry along with mirrorsymmetry can be found in Figure 4-11.Not only a symmetry plane but also other symmetry elements mayserve as antisymmetry elements.
We have already seen the contourof the oriental symbol Yin/Yang representing twofold rotationalFigure 4-10. Identity operation (top) and antiidentity operation (bottom).1984 Helpful Mathematical ToolsFigure 4-11. Mirror symmetries and antimirror symmetries: 1–2 and 3–4 mirrorsymmetries; 1–4 and 2–3 antimirror symmetries.symmetry in Figure 2-12a. The complete sign also has a black/whitecolor change and thus shows twofold antirotational symmetry:Beside color change this symbol represents a whole array of opposites, such as night/day, hot/cold, male/female, young/old, etc.Figure 4-12 illustrates different combinations of symmetryelements, for example, twofold, fourfold, and sixfold antirotationaxes together with other symmetry elements after Shubnikov [15].The fourfold antirotation axis includes a twofold rotation axis, andthe sixfold antirotation axis includes a threefold rotation axis.
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