M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 19
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Examples:Ih : The most characteristic feature of this point group is the presenceof six fivefold rotation axes. Examples:3.6. Consequences of SubstitutionA tetrahedral AX4 molecule, for example, methane, CH4 , has the pointgroup of the regular tetrahedron, Td . Gradual substitution of the Xligands by B ligands leads to less symmetrical tetrahedral configurations (Figure 3-12 top), until complete substitution is accomplished,where Td symmetry is re-established.
If each consecutive substitutionintroduces a new kind of ligand, then the symmetry will continueto decrease. This is shown for the tetrahedral case in the bottom ofFigure 3-12.Let us consider now an octahedral AX6 molecule, e.g., sulfurhexafluoride, SF6 , which has the symmetry of the regular octahedronOh . Substitution of an X ligand by a B ligand results in an AX5 B1163 Molecular Shape and GeometryFigure 3-12. Substitution in a tetrahedral AX4 molecule. Top: Gradual substitutionof the ligands X by ligands B; Bottom: Substitution of the ligands X by differentligands.molecule whose symmetry is C4v . The substitution of a second Xligand by another ligand B may lead to alternative structures as thesites of the five X ligands after the first substitution are no longerequivalent.
The symmetry variations in this substitution process areillustrated in Figure 3-13. A yet larger variety is obtained if eachconsecutive substitution introduces a new kind of ligand.Another example among fundamental structures is the benzenegeometry, D6h . Gradual substitution of an increasing number of hydrogens by ligands X results in the symmetry variations illustrated inFigure 3-14. As regards the molecular point group, the monosubstituted and the pentasubstituted derivatives are equivalent. All derivatives can be grouped in such pairs with each of the trisubstitutedbenzenes constituting a pair by itself. Again, only the simplest caseis considered here, with one kind of ligand used in all substitutedpositions.
The symmetry decrease in the molecular point group forthe substituted derivatives occurs because of the presence of thesubstituent ligands. It does not presuppose a change in the hexagonal symmetry of the benzene ring itself. Modern structure analyses have determined, however, that an appreciable deformationof the ring may also take place depending on the nature of thesubstituents. The largest deformation usually occurs at the so-calledipso angle adjacent to the substituent.
According to the general3.6. Consequences of Substitution117Figure 3-13. Gradual substitution of the ligands X in an octahedral AX6 moleculeby ligands B.observation, electronegative substituents tend to compress the ringwhile electropositive substituents elongate it [18].Complex formation usually implies the association of moleculesor other species which may also exist separately in chemically nonextreme conditions. Complex formation often has important consequences on the shapes and symmetries of the constituent molecules,determined also by the energy requirements of the geometricalchanges [19].
The H3 N·AlCl3 donor–acceptor complex, for example,has a triangular antiprismatic shape with C3v symmetry (Figure 3-15).The symmetry of the donor part (NH3 ) remains unchanged in thecomplex and the geometrical changes are relatively small. On theother hand, there are more drastic geometrical changes in the acceptorpart (AlCl3 ) due to loss of coplanarity of the four atoms and thisresults in a reduction in the point group. However, the structuralchanges in the acceptor part may also be viewed as if the complexformation completes the tetrahedral configuration around the centralatoms in the component molecules.
The nitrogen configuration maybe considered to be tetrahedral already in ammonia with the lonepair of electrons being the fourth ligand. For aluminum, it is indeedthe complexation that makes the tetrahedral configuration complete.1183 Molecular Shape and GeometryFigure 3-14. The symmetries of benzene and its C6 Hn X6-n derivatives.Coordination molecules often demonstrate the utility of polyhedra indescribing molecular shapes, symmetries, and geometries.
Of course,such description may be useful in many other classes of compoundsas well.Figure 3-15. The uncomplexed ammonia and aluminium trichloride molecules andthe triangular antiprismatic shape of the H3 N·AlCl3 donor–acceptor complex.3.7. Polyhedral Molecular Geometries1193.7. Polyhedral Molecular GeometriesIn the Preface to the Third Edition of his Regular Polytopes [20],the great geometer H. S.
M. Coxeter calls attention to the icosahedral structure of a boron compound in which twelve boron atomsare arranged like the vertices of an icosahedron. It had been widelybelieved that there would be no inanimate occurrence of an icosahedron, or of a regular dodecahedron either.In 1982, the synthesis and properties of a new polycyclic C20 H20hydrocarbon, dodecahedrane, was reported [21]. The twenty carbonatoms of this molecule are arranged like the vertices of a regulardodecahedron.
When, in the early 1960s, H. P. Schultz discussed thetopology of the polyhedrane and prismane molecules (vide infra) [22],at that time it was in terms of a geometrical diversion rather than truelife chemistry. Since then it has become real chemistry.It should be reemphasized that the above high-symmetry examplesrefer to isolated molecules and not to crystal structures. Crystallography has, of course, been one of the main domains where the importance of polyhedra has been long recognized, but they are not lessimportant in the world of molecules.In the First Edition of Regular Polytopes, Coxeter stated, “...the chief reason for studying regular polyhedra is still the same asin the times of the Pythagoreans, namely, that their symmetricalshapes appeal to one’s artistic sense” [23].
The success of modernmolecular chemistry does not diminish the validity of this statement. On the contrary. There is no doubt that aesthetic appeal hasmuch contributed to the rapid development of what could be termedpolyhedral chemistry. One of the pioneers in the area of polyhedralborane chemistry, Earl Muetterties, movingly described his attractionto the chemistry of boron hydrides, comparing it to M. C.
Escher’sdevotion to periodic drawings [24]. Muetterties’ words are quotedhere [25]:When I retrace my early attraction to boron hydridechemistry, Escher’s poetic introspections strike afamiliar note. As a student intrigued by earlydescriptions of the extraordinary hydrides, I hadnot the prescience to see the future synthesisdevelopments nor did I have then a scientific1203 Molecular Shape and Geometryappreciation of symmetry, symmetry operations,and group theory. Nevertheless, some inner forcealso seemed to drive me but in the direction ofboron hydride chemistry.
In my initial synthesisefforts, I was not the master of these molecules;they seemed to have destinies unperturbed by mythen amateurish tactics. Later as the developmentsin polyhedral borane chemistry were evident on thehorizon, I found my general outlook changed in acharacteristic fashion.
For example, my doodling,an inevitable activity of mine during meetings,changed from characters of nondescript form topolyhedra, fused polyhedra and graphs.I (and others, my own discoveries were notunique nor were they the first) was profoundlyimpressed by the ubiquitous character of the threecenter relationship in bonding (e.g., the boranes)and nonbonding situations. I found a singularuniformity in geometric relationships throughoutorganic, inorganic, and organometallic chemistry:The favored geometry in coordination compounds,boron hydrides, and metal clusters is the polyhedron that has all faces equilateral or near equilateral triangles...∗The polyhedral description of molecular geometries is, of course,generally applicable as these geometries are spatial constructions.
Toemphasize that even planar or linear molecules are also included,the term polytopal could be used rather than polyhedral. The realutility of the polyhedral description is for molecules possessing acertain amount of symmetry. Because of this and also because of theintroductory character of our discussion, only molecules with relatively high symmetries will be mentioned, but involving quite diverseexamples.Both the tetraarsene, As4 , and the methane, CH4 , molecules havetetrahedral shapes (Figure 3-16) and Td symmetry. However, there isan important difference in their structures.
In the As4 molecule, all∗Reproduced by permission from Academic Press.3.7. Polyhedral Molecular Geometries121Figure 3-16. The molecular shapes of As4 and CH4 .the four constituting nuclei are located at the vertices of a regulartetrahedron, and all the edges of this tetrahedron are chemical bondsbetween the As atoms. In the methane molecule, there is a centralcarbon atom, and four chemical bonds are directed from it to the fourvertices of a regular tetrahedron where the four protons are located.The edges are not chemical bonds.The As4 and CH4 molecules are clear-cut examples of the twodistinctly different arrangements.
However, these distinctions are notalways so unambiguous. Two independent studies reported the structure of zirconium borohydride, Zr(BH4 )4 . Both described its geometry by the same polyhedral configuration, while they differed in theassignment of the chemical bonds (Figure 3-17). The most importantFigure 3-17. The molecular configuration of zirconium borohydride, Zr(BH4 )4 , intwo interpretations but described by the same polyhedral shape.
Left: the zirconiumatom is directly bonded to the four tetrahedrally arranged boron atoms [28]; Right:the zirconium and the tetrahedrally arranged boron atoms are not bonded directly;their linkage is established by four times three hydrogen bridges [29].1223 Molecular Shape and Geometrydifference in the two interpretations concerned the linkage betweenthe central zirconium atom and the four boron atoms situated in thefour vertices of a regular tetrahedron.
According to one interpretation [20], there are four Zr–B bonds in the tetrahedral arrangement,while according to the other [27], there is no direct Zr–B bond, buteach boron atom is linked to the zirconium atom by three hydrogenbridges.The discovery of buckminsterfullerene (see, also, in Chapter 1) withits intriguing shape, focused attention to polyhedral molecular geometries even by many outside of chemistry. The event is often considered to be the birth of nanoscience and nanotechnology although theyexisted before even though under less fancy names. Buckminsterfullerene, C60 , discovered in 1985 [30], was the first runner-up forthe title, “Molecule of the Year” in 1990 [31], and received the titlein 1991 [32].
On this occasion, the Editorial of Science stated, amongothers, thatPart of the exhilaration of the fullerenes is theshock that an old reliable friend, the carbonatom, has for all these years been hiding asecret life-style. We were all familiar with thecharming versatility of carbon, the backbone oforganic chemistry, and its infinite variation inaromatic and aliphatic chemistry, but when yougot it naked, we believed it existed in two wellknown forms, diamond and graphite. The findingthat it could exist in a shockingly new structureunleashes tantalizing new experimental and theoretical ideas [33].Then it added something that certainly carried a flavor of thebroadest possible implications: “Perhaps the least surprising mightbe that improving life through science is a path that would see allthe citizens of the world holding hands like carbon atoms in C60 andlike them, welcoming any newcomer, no matter how different his orher skills or challenges.” Figure 3-18 shows a series of fullerenes,the C20 molecule of the shape of dodecahedron being the smallest.Due to its extreme curvature and supposed reactivity, its existence hadbeen in doubt until 2000, when it was produced from dodecahedrane,C20 H20 [34].3.7.
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