M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 18
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The Schoenflies notation is used and the characteristicsymmetry elements are enumerated.C1 : There are no symmetry elements except the one-fold rotationaxis, or identity, of course. C1 symmetry is asymmetry. Examples are:C2 , C3 , C4 , C5 , C6 , .
. . , Cn : One twofold, threefold, fourfold, fivefold, sixfold rotation axis, respectively, and it can be continued byanalogy. Cn has one n-fold rotation axis. Examples: Figure 3-6a. Themost famous molecule that has C2 symmetry is deoxyribonucleic acid(DNA) whose double helical structure will be discussed in more detailin Chapter 8. Here, suffice it to note that the C2 symmetry “wouldmake a model of DNA suitable for use as a staircase in a space ship”because “these elements are twofold rotation axes passing througheach of the base pairs at right angles to the helical axis; each of thembrings one of the chains into congruence with its partner of opposite polarity by a rotation of 180◦ ” [11].
The C2 symmetry playedan important role in the discovery of the double helix and is intimately related to the genetic function of the molecule [12]. Anotherimportant biological system that also has C2 symmetry, even thoughin an approximate way only, is the photosynthetic reaction center(Figure 3-6b). Whereas the C2 symmetry of DNA has well-definedfunctional implications, no such meaning of this symmetry for theprocess of photosynthesis has been uncovered (yet?) [13]. This is how1083 Molecular Shape and Geometrychemistry Nobel laureate Johann Deisenhofer who first noticed thissymmetry described the moment as he was locating the chlorophyllmolecules in the structure [14]:It was extremely exciting to localize these featuresand build models for them. When I stepped backto see the arrangement, the unexpected observation about it was symmetric. There was a symmetryin the arrangement of the chlorophyll that nobodyhad anticipated.
Nobody, to this day, completely(a)(b)Figure 3-6. (a) Molecules illustrating Cn symmetries; (b) The structure of thephotosynthetic reaction center with approximate C2 symmetry (courtesy of JohannDeisenhofer, Dallas, Texas) [15].3.5. Examples109understands the purpose of this symmetry. I thinkit can be understood only on the basis of evolution.I think that the photosynthetic reaction started outas a totally symmetric molecule.
Then it turned outto be preferable to disturb its symmetry, sticking toan approximate symmetry but changing subtly thetwo halves of the molecule. Because of the difference in properties of the two halves, the conclusion had been, before the structure came out, thatthere cannot be symmetry; that it has to be anasymmetric molecule. Now when people lookedat the structure, it looked totally symmetric to thenaked eye. That realization was the high point Iwill never forget.Deisenhofer’s description is a beautiful illustration for some of theideas about the importance of symmetries occurring in an approximate way as discussed in the Introduction (Chapter 1).
The near-C2symmetry of the photosynthetic reaction center [16] and its elucidation [17] have been discussed in the literature.Ci : Center of symmetry. Examples:Cs : One symmetry plane. Examples:1103 Molecular Shape and GeometryS4 : One fourfold mirror-rotation axis.S6 : One sixfold mirror-rotation axis, which is, of course, equivalent toone threefold rotation axis plus center of symmetry. Example:C2h , C3h , ..., Cnh : One twofold, threefold, ..., n-fold rotation axis witha perpendicular to it symmetry plane. Examples: Figure 3-7.Figure 3-7. Examples with rotational axis and perpendicular symmetry plane, Cnh .C2v , C3v , C4v , C5v , C6v , ..., Cnv : C2v , Two perpendicular symmetryplanes whose crossing line is a twofold rotation axis; C3v , One threefold rotation axis with three symmetry planes which include the rotation axis.
The angle is 60◦ between two symmetry planes; C4v , Onefourfold rotation axis with four symmetry planes which include therotation axis. The four planes are grouped in two nonequivalent pairs.One pair is rotated relative to the other pair by 45◦ . The angle between3.5. Examples111the two planes within each pair is 90◦ . This series can be continuedby analogy. When n is even, there are two sets of symmetry planes.One set is rotated relative to the other set by (180/n)◦ .
The anglebetween the planes within each set is (360/n)◦ . When n is odd, theangle between the symmetry planes is (180/n)◦ . Examples: Figure 3-8.C∞v : One infinite-fold rotation axis with infı̀nite number of symmetryplanes which include the rotation axis. Example: Figure 3-8.Figure 3-8. Examples with rotation axis and symmetry planes containing the rotation axis, Cnv .D2 : Three mutually perpendicular twofold rotation axes.D3 : One threefold rotation axis and three twofold rotation axesperpendicular to the threefold axis.
The twofold axes are at 120◦ , sothe minimum angle between two such axes is 60◦ Examples:1123 Molecular Shape and GeometryD4 : One fourfold rotation axis and four twofold rotation axes whichare perpendicular to the fourfold axis. The four axes are grouped intwo nonequivalent pairs. One pair is rotated relative to the other pairby 45◦ . The angle between the two axes within each pair is 90◦ .D5 , D6 , D7 , ..., Dn : This series can be continued by analogy.
It ischaracterized by one n-fold rotation axis and n twofold rotation axesperpendicular to the n-fold axis.D2d : Three mutually perpendicular twofold rotation axes and twosymmetry planes. The planes include one of the three rotation axesand bisect the angle between the other two. Example: Figure 3-9.Figure 3-9. Dnd symmetries.D3d : One threefold rotation axis with three twofold rotation axesperpendicular to it, and three symmetry planes.
The angle between thetwofold axes is 60◦ . The symmetry planes include the threefold axisand bisect the angles between the twofold axes. Examples: Figure 3-9.D4d , D5d , D6d , D7d , . . ., Dnd : D4d One fourfold rotation axis with fourtwofold rotation axes perpendicular to it, and four symmetry planes.The angle between the twofold axes is 45◦ . The symmetry planesinclude the fourfold axis and bisect the angles between the twofoldaxes.
The series can be continued by analogy. Examples: Figure 3-9.3.5. Examples113D2h : Three mutually perpendicular symmetry planes. Their threecrossing lines are three twofold rotation axes, and their crossing pointis a center of symmetry. Examples: Figure 3-10.Figure 3-10. Dnh symmetries.D3h : One threefold rotation axis, three symmetry planes (at 60◦ ) whichcontain the threefold axis, and another symmetry plane perpendicularto the threefold axis. Examples: Figure 3-10.D4h : One fourfold axis, one symmetry plane perpendicular to it, andfour symmetry planes which include the fourfold axis.
The four planesmake two pairs. One pair is rotated relative to the other pair by 45◦ .The two planes in each pair are perpendicular to each other. Example:Figure 3-10.D5h : One fivefold rotation axis, one symmetry plane perpendicularto it, and five symmetry planes which include the fivefold rotationaxis. The angle between the adjacent five planes is 36◦ . Example:Figure 3-10.D6h : One sixfold rotation axis, one symmetry plane perpendicular toit, and six symmetry planes which include the sixfold axis.
The sixplanes are grouped in two sets. One set is rotated relative to the other1143 Molecular Shape and Geometryset by 30◦ . The angle between the planes within each set is 60◦ . Examples: Figure 3-10.Dnh : The series can be continued by analogy. There will be onen-fold rotation axis, one symmetry plane perpendicular to it, and nsymmetry planes which include the n-fold axis. When n is even, thereare two sets of symmetry planes. One set is rotated relative to theother set by (180/n)◦ . The angle between the planes within each setis (360/n)◦ .
When n is odd, the angle between the symmetry planesis (180/n)◦ .D∞h : One ∞-fold axis and a symmetry plane perpendicular to it. Ofcourse, there are also ∞ number of symmetry planes which includethe ∞-fold rotation axis. Example: Figure 3-10.T: Three mutually perpendicular twofold rotation axes and four threefold rotation axes. The threefold axes all go through a vertex ofa tetrahedron and the midpoint of the opposite face center. Thetwofold axes connect the midpoints of opposite edges of this tetrahedron.
Example: Figure 3-11.Figure 3-11. T symmetries.Td : In addition to the symmetry elements of symmetry T, there aresix symmetry planes, each two of them being mutually perpendicular.All of these symmetry planes contain two threefold axes. Examples:Figure 3-11.3.6. Consequences of Substitution115Th : In addition to the symmetry elements of symmetry T, there isa center of symmetry, which introduces also three symmetry planesperpendicular to the twofold axes. Example: Figure 3-11.Oh : Three mutually perpendicular fourfold rotation axes and fourthreefold rotation axes which are tilted with respect to the fourfoldaxes in a uniform manner, and a center of symmetry.
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