Establishing the Point Group

Hermann-Mauguin Schoenflies Hermann-Mauguin Schoenflies 

Point-group symmetries not listed in Table 3-1 may easily be assigned the appropriate Schoenflies notation by analogy. Thus, e.g., C5v, C5h, C7, Cg, etc. can be established. Such symmetries may well occur among real molecules. 

If the molecule does not belong to one of these special groups, a systematic approach is followed. Firstly, the possible presence of rotation axes in the molecule is checked. If there is no rotation axis, then it is determined whether there is a symmetry plane (Cs). In the absence of rotational axes and mirror planes, there may only be a center of symmetry (C,), or there may be no symmetry element at all (Ci). If the molecule has rotation axes, it may have a mirror-rotation axis with even-number order (S2n) coinciding with the rotation axis. For. S4 there will be a coinciding C2, for S6 a coinciding C3, and for S%, both C2 and C4. 

In any case the search is for the highest order C axis. Then it is ascertained whether there are n C2 axes present perpendicular to the C axis. If such C2 axes are present, then there is D symmetry. If in addition to D symmetry there is a a plane, the point group is Dnh, while if there are n symmetry planes (arf) bisecting the twofold axes, 

Finally, if no C2 axes perpendicular to C are present, then the lowest symmetry will be C , when a perpendicular symmetry plane is present, it will be Cnh, and when there are n coinciding symmetry planes, the point group will be Cm. 

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In the HMBC spectrum this NH proton exhibited 3/(CH) correlation to the C3" (5174.15) and 2/(CH) interaction to the C13 (5165.48) carbonyl, the latter in turn showing 2/(CH) interaction with H12(56.05). Thus it became clear that a carboxamide group linked the conjugated triene and the cyclopentenone unit. A 2/(CH) coupling of the triene terminus H7 to C4 (5jj/5c 5.86/71.20) established the point of attachment of the triene unit to C4. An observed NOE interaction between the H7 and H5 lent further support to this attachment and was suggestive of proximal orientation of the trans-A1 bond to the epoxy unit in the most preferred conformation. The absolute configuration at C4 was not established. 

The following simple rules can be established for point groups ... 

A fundamental property of the wave function is that it can be used as basis for irreducible representations of the point group of a molecule [13], This property establishes the connection between the symmetry of a molecule and its wave function. The preceding statement follows from Wigner s theorem, which says that all eigenfunctions of a molecular system belong to one of the symmetry species of the group [14],... 

In Fig. 7-9.1 the symmetry elements for the point group are shown (see also Fig. 3-6.1) as well as our choice of x, y and z axes (this ohoioe establishes the orientation of the p- and d-orbitals). In Table 7-9.1 the csorresponding character table is given in full. 

Polar alignment resulting from this process establishes C ov symmetry in the medium. There are two unique directions in the point group, the polar direction X and the direction perpendicular and cylindrically symmetric with respect to the X direction. When the common assumption that frequencies and the Cartesian axes along which they are applied can be freely interchanged in lossless media is invoked, only two nonequivalent tensor components exist. 

Dehydrogenation of the lactone acid (XVII) with palladium-charcoal yielded 2-ethylnaphthalene and under the same conditions the acid X afforded 2-ethyl-3-methylnaphthalene. The formation of these two hydrocarbons disproved the earlier hypotheses that the carbonyl group of himbacine was attached to a tertiary carbon atom and that the ethyl groups of 2-ethyl-6-methylpyridine and 2-ethyl-3-methylnaphthalene, isolated from the selenium dehydrogenation of himbacine, were derived from the same structure in the alkaloid. Moreover, since the results of the Hofmann degradation and the oxidation experiments had shown that the 6-(l,2-dimethylpiperidyl)-vinyl group was attached to a decalin residue by reason of the formation of the C-13 ketones, it only remained to establish the point of attachment. For this purpose the carboxyl group... 

If the point group symmetry is equivalent to orthorhombic or lower, i.e. if the optical properties are biaxial, the local extinction directions are not simply related to the director as projected onto the specimen plane. It may be possible to use light microscopy to establish whether or not a region of the specimen is optically biaxial ... 

Consider the complex [Ni(CO)3] in this geometry. By making use of the results established in Chapter 6 ( 6.6.6.2), give the symmetry-adapted linear combinations of the orbitals (the point group is Dsh, whose character table is given in Table 6.21). 

A set of symmetry elements (or of symmetry operations) enables us to define a symmetry point group which is represented by a symbol. There is a simple method to determine the point group, and it is not necessary to establish a complete list of all the symmetry operations in the molecule under consideration. 

The Cgo structure, the most stable of this family, is a perfect truncated icosahedron, a geometric solid having 20 6-membered rings and the considerable symmetry of the point group I), 6 C5 axes, 10 C3 axes, and 15 Q axes. The symmetry was first established by measuring the C NMR spectrum of in which only one peak was observed, verifying that all 60 carbon atoms were chemically equivalent. 

A 1 2 complex between Ni " " and N,N-dimethyl-/3-mercaptoethylamine, Me2NCH2CH2SH, has been synthesized. It is desired to establish the configuration (cis or trans) of the complex and the coordination (square planar or tetrahedral) of Ni +. Explain how the following information enables one to elucidate the stereochemistry. Also state the point group to which it belongs and sketch the bis(N,JV-dimethyl- 3-mercaptoethylamine)nickel(ll) molecule. 

Matrix blocking will already be effected by the point-group symmetry of the molecule. We do not expect time reversal to reduce the symmetry because the time-reversal operator K commutes with all operations of the point group. If time reversal introduces new symmetries, we will gain further blocking. We must therefore establish the relation between time-reversal symmetry and point-group symmetry. 

In this chapter we will explore further the symmetry operations that are used to describe molecular structure. New operations are introduced to complete the set used in molecular symmetry. Particular sets of operations often recur, with many molecules having the same collection of operations. Once we establish how the properties of a molecule depend on the set of valid operations, this will mean that we can actually infer the properties of many related molecules of the same symmetry. The sets of operations are referred to as point groups, and our main task in this chapter is to introduce the concept of a point group. The formal construction of the point groups most commonly used in chemistry is carried out in Chapter 3. The properties of the point groups, and their application in vibrational spectroscopy and MO theory, are then the subject of the remainder of the book. 

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