M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 17
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On the other hand, three hydrogens are equivalent in bothmethyl formate and acetic acid, with the fourth being different in thetwo molecules. There are three different types of hydrogen positionsin glycol aldehyde.Molecules are structural isomers if they have the same empiricalformula but the distances between corresponding atoms are not thesame (Figure 3-1).
Structural isomers are of two types. If their atomicconnectivities are the same, they are diastereomers, and if their atomicconnectivities are different, they are constitutional isomers. Somediastereomers become superimposable by rotation about a bond, andthey are called rotational isomers. Depending on the magnitude of the1003 Molecular Shape and GeometryFigure 3-1. The hierarchy of isomers.barrier to rotation, geometrical isomers (high barrier) and conformers(low barrier) are distinguished.Identical molecules have the same formula, the same atomicconnectivity, and the same distances between corresponding atoms.In addition, they are superimposable (homomers).
Enantiomers havethe same formula, the same atomic connectivity, and the samedistances between corresponding atoms, but they are not superimposable, instead, they are mirror images of each other (cf. Section 2.7 onchirality).3.2. Rotational IsomerismThe four-atomic chain is the simplest system for which rotationalisomers are possible, as shown in Figure 3-2. Rotational isomers,or conformers, are various forms of the same molecule related by3.2. Rotational Isomerism101Figure 3-2. Rotational isomerism of a four-atomic chain.rotation around a bond as axis. The rotational isomers of a moleculeare described by the same empirical formula and by the samestructural formula. Only the relative positions of the two bonds (orgroups of atoms) at the two ends of the rotation axis are changed.The molecular point groups for different rotational isomers may beentirely different.Rotational isomers can be conveniently represented by so-calledprojection formulae in which the two bonds (or groups of atoms)at the two ends are projected onto a plane which is perpendicularto the central bond.
This plane is denoted by a circle whose centercoincides with the projection of the rotation axis. The bonds infront of this plane are drawn as originating from the center, whilethe bonds behind this plane, i.e., the bonds from the other end ofthe rotation axis, are drawn as originating from the perimeter of thecircle.The drawings by Degas End of the Arabesque and Seated DancerAdjusting Her Shoes may be looked at as illustrations of the staggered and eclipsed conformations of the molecule A2 B–BC2 [5].The dancers and their projection-like representations are depictedin Figure 3-3 along with the projectional representations of twoconformers of the molecule A2 B–BC2 .
The projections in Figure 3-3represent views along the B–B bond, i.e., the dancer’s body. The planebisecting the B–B bond is shown by the circle and it corresponds to thedancer’s skirt. The dancer’s arms and legs refer to the bonds B–A andB–C, respectively. Incidentally, the bouquet in the right hand of thedancer in the staggered conformation might be viewed as a differentsubstituent.Two important cases in rotational isomerism are distinguished byconsidering the nature of the central bond. When it is a double bond,rotation of one form into another is hindered by a high potential1023 Molecular Shape and Geometrybarrier.
This barrier may be so high that the two rotational isomerswill be stable enough to make their physical separation possible. Anexample is 1,2-dichloroethylene.The symmetry of the cis isomer is characterized by two mutuallyperpendicular mirror planes generating also a two-fold rotational axis.This symmetry class is labeled mm. An equivalent notation is C2v asFigure 3-3. Projectional representation of rotational isomers [6]; Top, left: drawingafter Degas’ End of the Arabesque by Ferenc Lantos; Right: drawing after Degas’Seated Dancer Adjusting Her Shoes by Ferenc Lantos; Middle: contour drawingsof the dancers; Bottom: staggered and eclipsed rotational isomers of the A2 BBC2molecule by projections representing view along the B–B bond.3.2.
Rotational Isomerism103will be seen in the next section. The trans isomer has one twofoldrotation axis with a perpendicular symmetry plane, its symmetry classis 2/m (C2h ).Rotational isomerism relative to a single bond is illustrated byethane and 1,2-dichloroethane, both depicted in Figure 3-4. First, takethe ethane molecule, H3 C–CH3 . During a complete rotation of onemethyl group around the C–C bond relative to the other methyl group,(a)(b)Figure 3-4.
Potential energy functions for rotation about a single bond, isthe angle of rotation. (a) Ethane, H3 C–CH3 . There are two different symmetricalforms. Both the staggered form with D3d symmetry and the eclipsed form with D3hsymmetry occur three times in a complete rotational circuit; (b) 1,2-dichloroethane,ClH2 C–CH2 Cl. There is no other symmetrical form in the region between the twosymmetrical staggered forms shown. The eclipsed form with C2v symmetry and thestaggered form with C2h symmetry occur once, while the staggered form with C2symmetry occurs twice in a complete rotational circuit.1043 Molecular Shape and Geometrythe ethane molecule appears three times in the stable staggered formand three times in the unstable eclipsed form. As all the hydrogenatoms of one methyl group are equivalent, the three energy minima areequivalent, and so are the three energy maxima, as seen in Figure 3-4a.The situation becomes more complicated when the three ligandsbonded to the carbon atoms are not the same.
This is seen for 1,2dichloroethane in Figure 3-4b. There are three highly symmetricalforms. Of these two are staggered with C2h and C2 symmetries, respectively. The third is an eclipsed form with C2v symmetry. This form hasCl/Cl and H/H eclipsing.Figure 3-4 shows only the symmetrical conformers by projectionformulae.
The symmetrical forms always belong to extreme energies, either minima or maxima. The barriers to internal rotation in thepotential energy functions depicted in Figure 3-4 are about 10 kJ/mol.Typical barriers for systems where the double bonds would be considered to be the “rotational axis” may be as much as 30 times greaterthan those for systems with single bonds.3.3.
Symmetry NotationsSo far, the so-called International or Hermann-Mauguin symmetrynotations have been used in the descriptions in this text. Another,older system by Schoenflies is generally used, however, to describethe molecular point-group symmetries. This notation has been givenin parenthesis in the preceding section. The Schoenflies notation hasthe advantage of succinct expression for even complicated symmetryclasses combining various symmetry elements.
The two systems arecompiled in Table 3-1 [7] for a selected set of symmetry classes.The set includes all point-group symmetries in the world of crystals which are restricted to 32 classes. The reasons and significanceof these restrictions will be discussed later in the chapter on crystals(Section 9.3). There are no restrictions on the point-group symmetriesfor individual molecules, and a few further, so-called limiting, classesare also listed in Table 3-1.The Schoenflies notation for rotation axes is Cn , and for mirrorrotation axes the notation is S2n , where n is the order of the rotation.
The symbol i refers to the center of symmetry (cf. Section 2.4).Symmetry planes are labeled ; v is a vertical plane, which alwayscoincides with the rotation axis with an order of two or higher, and3.4. Establishing the Point Group105Table 3-1. Symmetry Notations of the Crystallographic and a Few Limiting GroupsHermann-MauguinSchoenfliesHermann-MauguinSchoenfliesCrystallographic groups11̄m22/mmm222mmm44̄4/m4mm4̄2m4224/mmm33̄3m32C1CiCsC2C2hC2vD2D2hC4S4C4hC4vD2dD4D4hC3S6C3vD33̄m6̄66/m6̄m26mm6226/mmm23m3̄4̄3m432m3mLimiting groups∞∞2∞/m∞mm∞/mmD3dC3hC6C6hD3hC6vD6D6hTThTdOOhC∞D∞C∞hC∞vD∞hh is a horizontal plane, which is always perpendicular to the rotationaxis when it has an order of two or higher.Point-group symmetries not listed in Table 3-1 may easily beassigned the appropriate Schoenflies notation by analogy.
Thus, e.g.,C5v , C5h , C7 , C8 , etc. can be established. Such symmetries may welloccur among real molecules.These systems of notation have been well established and widelyused. Nonetheless, other systems might be and have been suggestedthough none has gone into practice. We mention here one such suggestion by outstanding mathematicians whose system has merits, but onlytime will tell whether it might gain acceptance [8].3.4. Establishing the Point GroupFigure 3-5 shows a possible scheme for establishing the molecularpoint group that has been widely used to reliably establish molecularsymmetries [9].1063 Molecular Shape and GeometryFigure 3-5.
Scheme for establishing the molecular point groups [10].First, an examination is carried out whether the molecule belongs tosome “special” group. If the molecule is linear, it may have a perpendicular symmetry plane (D∞h ) or it may not have one (C∞v ). Very highsymmetries are easy to recognize. Each of the groups T, Th , Td , O, andOh , has four threefold rotation axes. Both icosahedral I and Ih groupsrequire ten threefold rotation axes and six fivefold rotation axes.
Themolecules belonging to these groups have a central tetrahedron, octahedron, cube, or icosahedron.If the molecule does not belong to one of these “special” groups, asystematic approach is followed. Firstly, the possible presence of rotation axes in the molecule is checked. If there is no rotation axis, then itis determined whether there is a symmetry plane (Cs ).
In the absenceof rotational axes and mirror planes, there may only be a center ofsymmetry (Ci ), or there may be no symmetry element at all (C1 ). Ifthe molecule has rotation axes, it may have a mirror-rotation axis witheven-number order (S2n ) coinciding with the rotation axis.
For S4 therewill be a coinciding C2 , for S6 a coinciding C3 , and for S8 , both C2and C4 .In any case the search is for the highest order Cn axis. Then it isascertained whether there are n C2 axes present perpendicular to theCn axis. If such C2 axes are present, then there is D symmetry. If inaddition to D symmetry there is a n plane, the point group is Dnh ,while if there are n symmetry planes (d ) bisecting the twofold axes,3.5. Examples107the point group is Dnd . If there are no symmetry planes in a moleculewith D symmetry, the point group is Dn .Finally, if no C2 axes perpendicular to Cn are present, then thelowest symmetry will be Cn , when a perpendicular symmetry planeis present, it will be Cnh , and when there are n coinciding symmetryplanes, the point group will be Cnv .3.5. ExamplesIn this section, actual molecular structures are shown for the variouspoint groups.