M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 32
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Theantisymmetry elements have the same notation as the ordinary onesexcept that they are underlined. Antimirror rotation axes characterizethe rosettes in the second row of Figure 4-12. The antirotation axesappear in combination with one or more symmetry planes perpendicular to the plane of the drawing in the third row of Figure 4-12.Finally, the ordinary rotation axes are combined with one or moreantisymmetry planes in the two bottom rows of Figure 4-12. In fact,symmetry 1 · m here is the symmetry illustrated in Figure 4-11. Theblack-and-white variation is the simplest case of color symmetry.4.6.
Antisymmetry199Figure 4-12. Antisymmetry operations. First row: antirotation axes 2, 4, 6; Secondrow: antimirror rotation axes 2, 4, 6; Third row: antirotation axes combined withordinary mirror planes 2·m, 4·m, 6·m; Fourth row: ordinary rotation axes combinedwith antimirror planes 2 · m, 4 · m, 6 · m; Fifth row: 1 · m, 3 · m, after Shubnikov[16]. Reproduced with permission from Nauka Publishing Co., Moscow.2004 Helpful Mathematical ToolsFigure 4-13. Ornament on a building in Spain with 4 · m symmetry.
Photograph bythe authors.These considerations become more and more complicated withincreasing the number of colors [16–19]. Figure 4-13 shows anexample of 4 · m symmetry. The detail of the tower of a gatehouse atPark Güell (see Figure 4-14), the famous park in Barcelona built byAntoni Gaudi also reveals 4 · m symmetry, that is, fourfold rotationalsymmetry combined with antireflection.All the above examples applied to point groups.
Antisymmetry andcolor symmetry, of course, may be introduced in space-group symmetries as well as examples illustrate in Figures 8-32, 8-37, and 9-46(in the discussion of space groups). If we look only at the close-upof the tower in Figure 4-14b, it also has tranlational antisymmetry,specifically anti-glide-reflection symmetry together with similaritysymmetry (these symmetries will be discussed in Chapter 8).Color change is perhaps the simplest version of antisymmetry.
Thegeneral definition of antisymmetry, at the beginning of this section,however, calls for a much broader interpretation and application. Therelationship between matter and antimatter is a conspicuous exampleof antisymmetry. There is no limit to down-to-earth examples, as wellas to abstract ones, especially if, again, symmetry is considered ratherloosely.Figure 4-15 is another example of antimirror symmetry involvingcolor change. However, there is more than geometrical correspondence in this Soviet poster from 1987. The text says “This is perestroika to some,” implying dissatisfaction with the way reforms were4.6.
Antisymmetry(a)201(b)Figure 4-14. (a) The top of the tower of a gatehouse in Park Güell by AntoniGaudi in Barcelona, Spain, with 4·m symmetry; (b) Close-up of the tower, revealingtranslational antisymmetry together with similarity symmetry. Photograph by theauthors.Figure 4-15. Soviet (1987) poster on perestroika. Photograph by the authors.2024 Helpful Mathematical Toolsbeing carried out, amounting to mere color changes rather than∗substantial ones.Figure 4-16a shows the logo of a sporting goods store in Boston,Massachusetts.
Geometrical correspondence is gone, yet we have nodifficulty in recognizing the antimirror symmetry relationship. Theantireflection plane relates a half-snowflake and a half-sun, symbolizing winter and summer, respectively. There are two coke machinesin the picture of Figure 4-16b. There is no geometrical correspondence, but there is color reversal, and reversal of yet another, moreimportant, property, the sugar content.
This makes the two machinesan example of antisymmetry with some abstraction.Two old buildings with modern skyscrapers in the background andthe houses of a medieval Italian town with a radar locator in thebackground express the antisymmetric relationship of old and new(Figure 4-17), while the façade of the Notre Dame cathedral showingan angel and the devil expresses the antisymmetry between good andevil (Figure 4-18).The above examples of antisymmetry may have implied at leastas much abstraction as any chemical application.
The symmetric andFigure 4-16. (a) Logo of a sporting goods store in Boston, Massachusetts; (b) Twocoke machines where color change and, even more important, reversal of sugarcontent, make the antisymmetric relationship. Photographs by the authors.∗The Russian word “perestroika” means restructuring.4.6. Antisymmetry203Figure 4-17. Buildings in (a) Boston; (b) New York City; (c) Old buildings in Erice,Sicily, with a radar locator in the background.
They all illustrate the antisymmetrybetween new and old. Photographs by the authors.antisymmetric behavior of orbitals describing electronic structure, andvectors describing molecular vibrations may be perceived with greaterease after the preceding diversion.
Before that, however, some moreof group theory will be covered.Figure 4-18. Façade of the Notre Dame cathedral in Paris illustrating the antisymmetry between good and evil. Photograph by the authors.2044 Helpful Mathematical Tools4.7. Shortcut to Determine a RepresentationIt was quite easy to find the irreducible representation of Rz before, asthe representation we worked out appeared to be an irreducible representation itself. In most cases, however, a reducible representation isfound when the symmetry operations are applied to a certain basis.Now a simpler way will be shown (1) to describe the representationon a given basis without generating the matrices themselves and (2)to reduce them, if reducible, to irreducible representations.The diimide molecule (4-1) is our example again, and the basisis the two N–H bond length changes (see Figure 4-7).
It is easy togenerate the matrices corresponding to each operation using such asimple basis; however, even this may not be necessary. As mentionedbefore, instead of the representations themselves, we can work withtheir characters. For this particular case the characters of the representation have already been determined:⌫12002But how can we know the character of a matrix without writingdown the whole matrix?Looking back at the effect of the different symmetry operations on HNNH (Figure 4-7) it is recalled, for example, that C2interchanges ⌬r1 and ⌬r2 , so the diagonal elements of the matrixwill all be 0.
Consequently, these vectors do not contribute to thecharacter.This observation can be generalized: those basis elements that areassociated with an atom changing its position during the symmetryoperation will have zero contribution to the character. The basiselement that is unchanged by a given operation contributes +1 to thecharacter. Finally, the basis element that is transformed into its negative contributes −1. The only complication arises with the rotationaloperations when the atom does not move during the symmetry operation but the basis element associated with it is rotated by a certainangle. Here the matrix of the rotation has to be constructed as shownin Section 4.2.Returning to the diimide N–H bond length changes, let us seehow the above simple rules work. The identity operation, E, leaves4.7.
Shortcut to Determine a Representation205the molecule unchanged, so the two vectors, ⌬r1 and ⌬r2 , will eachcontribute +1 to the character:1+1=2The effect of C2 has already been looked at. Its character is 0. Theeffect of the inversion operation is the same as that of C2 , so the character will be0+0=0Finally, operation h leaves the two bonds unchanged, so both ofthem contribute +1 to the character:1+1=2The result is the same as before:⌫12002Now, check the rules with a larger basis set, the Cartesian displacement coordinates of the atoms of HNNH (see Figure 4-8).
OperationE leaves all the 12 vectors unchanged, so its character will be 12. C2brings each atom into a different position so their vectors will alsobe shifted. This means that all vectors will have zero contribution tothe character. The same applies to the inversion operation. Finally, asalready worked out before, the horizontal reflection leaves all the xand y vectors unchanged and brings the four z vectors into their negative selves.
The result is8 + (−4) = 4The whole representation of the displacement vectors is:⌫212004Both representations we constructed here are reducible since thereare no 2- and 12-dimensional representations in the C2h character table(Table 4-7). The next question is how to reduce these representations.2064 Helpful Mathematical Tools4.8. Reducing a RepresentationIt was discussed before that the irreducible representations can beproduced from the reducible representations by suitable similaritytransformations.
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