Symmetry Criterion

Enantiotopic ligands and faces are not interchangeable by operation of a symmetry element of the first kind (Cn, simple axis of symmetry) but must be interchangeable by operation of a symmetry element of the second kind (cr, plane of symmetry i, center of symmetry or S , alternating axis of symmetry). (It follows that, since chiral molecules cannot contain a symmetry element of the second kind, there can be no enantiotopic ligands or faces in chiral molecules. Nor, for different reasons, can such ligands or faces occur in linear molecules, QJV or Aoh ) 

The symmetry planes (a) in molecules 30, 32, 34, 36, 38, Fig. 13 should be readily evident. It is possible to have both homotopic and enantiotopic ligands in the same set, as exemplified by the case of cyclobutanone (34) HA and HD are homotopic as are HB and Hc, HA is enantiotopic with HB and Hc HD is similarly enantiotopic with Hc and HB. The sets HAjB and HC D may be called equivalent (or homotopic) sets of enantiotopic hydrogen atoms. The unlabeled hydrogens at position 3, constitutionally distinct — see Section 3.4 — from those at C(2, 4), are homotopic with respect to each other. Enantiotopic ligands need not be attached to the same atom — viz. the case of mew-tartaric acid (32) and also the just-mentioned pair Ha, Hc [or Hb, Hd] in cyclobutanone. 

Symmetry elements of the second kind other than ct may generate enantiotopic ligands. Thus compound 42 in Fig. 16 (F and T are enantiomorphic, i.e. mirror- 

Enantiotopic faces (Fig. 15) are also related by a symmetry plane — e.g. the plane of the double bond in 40. The faces must not be interchangeable by operation of a symmetry axis, lest they be homotopic rather than enantiotopic. 

So far we have discussed groups which are enantiotopic by internal comparison. Groups may also be enantiotopic by external comparison, i.e. groups in two different molecules are enantiotopic if they are related by reflection symmetry. Clearly this can be so only if the two molecules themselves are enantiomeric corresponding 

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Plane of symmetry. If a plane can be placed in space such that for every atom of the molecule not in the plane there is an identical atom (which is to say, the same atomic number and isotope) on the other side of the plane at equal distance from it (i.e., a mirror image ), the molecule is said to possess a plane of symmetry. The Greek letter o is often used to represent both the plane of symmetry and the operation of mirror reflection that it performs. An example of a molecule possessing a plane of symmetry is methylcyclobutane, as illustrated in Figure B.l. Note that a planar molecule always has at least one ct, since tire plane of tire molecule satisfies the above symmetry criterion in a trivial way (the set of reflected atoms is the empty set). Note also that if we choose a Cartesian coordinate system in such a way tliat two of the Cartesian axes lie in the symmetry plane, say x and y, then for every atom found at position (x,y,z) where z there must be an identical atom at position (x,y,—z). 

In order to take advantage of the symmetry criterion for orbital interaction, it is necessary to have orbitals that are symmetry correct. The molecular orbitals obtained from atomic orbitals by the methods described in Sections 1.2 and 10.1 will sometimes be symmetry correct and sometimes not. 

The addition criterion tends to be confusing when applied to a molecule like ethylene where addition occurs at both ends of the double bond. The reader is advised, in such cases, either to use the symmetry criterion or to choose epoxidation as the test reaction for the addition criterion. For additional examples involving the heterotopic faces of not only olefins and carbonyl compounds... 

Type of groups Molecular environment Symmetry criterion Substitution with achiral or chiral test group yields Elements of prostereoisomerism... 

Scheme 37. The symmetry criterion for cation radical formation via outer sphere electron transfer. Substituent effects in mono- and disubstituted stilbenes are multiplicative log krei correlates with (t+ with p = -4.16 the oxidation potentials of these same stilbene derivatives correlate with the same substituent parameters with p = -5.02.

Symmetry and stability analysis. The semi-empirical unrestricted Hartree-Fock (UHF) method was used for symmetry and stability analysis of chemical reactions at early stage of our theoretical studies.1,2 The BS MOs for CT diradicals are also expanded in terms of composite donor and acceptor MOs to obtain the Mulliken CT theoretical explanations of their electronic structures. Instability in chemical bonds followed by the BS ab initio calculations is one of the useful approaches for elucidating electronic structures of active reaction intermediates and transition structures.2 The concept is also useful to characterize chemical reaction mechanisms in combination with the Woodward-Hoffman (WH) orbital symmetry criterion,3 as illustrated in Figure 1. According to the Woodward-Hoffmann rule,3 there are two types of organic reactions orbital-symmetry allowed and forbidden. On the other hand, the orbital instability condition is the other criterion for distinguishing between nonradical and diradical cases.2 The combination of the two criteria provides four different cases (i) allowed nonradical (AN), (ii) allowed radical (AR), (iii) forbidden nonradical (FN), and (iv) forbidden radical (FR). The charge and spin density populations obtained by the ab initio BS MO calculations are responsible for the above classifications as shown in Fig. 1. 

Symmetry equivalent internal modes associated with symmetry equivalent internal coordinates must have the same amplitudes in the case that the normal mode being decomposed is symmetric. Symmetry criterion)... 

The same symmetry criterion used before was applied to select the conformations for the torsional coordinates. In addition, five values of the CNC angle around the equilibium, were also selected for describing the variation of the energy with the bending. These angle values were -5.0",-3.0 . 0.0" 3.0" and 5.0". The relative energy values were fitted to the expression for the potential. 

The symmetry criterion used in the present chapter adopts enantiotopic and/or f 5-diastereotopic relationships after drawing stereoisograms for testifying prochirality and/or pro-f 5-stereogenicity (Fujita 2009c). 

The connection between the two approaches described above is also shown in Table A13.1. At the right-hand side of each character table is listed the number of nodal planes containing the z axis associated with each type of interaction. There is something to be said in favour of each model in comparison with the other. Thus, the number of nodal planes criterion does not discriminate between and B2 (in the Av and Qt cases), yet only one is involved in the bonding. Conversely, although in each case the number of nodal planes quoted is the lowest, for each case higher numbers exist also (we give these for the Ai case). In such cases, the number of nodal planes criterion provides a distinction that the symmetry criterion does not. Only orbitals with the same number of nodal planes interact. 

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