Reflection symmetry elements

In the third model (finite chain with different terminal groups) no reflection symmetry element exists in the Fischer projection. The individual macromolecules are, therefore, chiral and all the tertiary atoms are asymmetric and different. The stereochemical notation for a single chain, depending on the priority order of the end groups, can be R, R2, R. . . R -2, R -i, Rn or R, R2, R3... 

The key to the generalization achieved is then the removal of the reflection symmetry elements in the space and time coordinates in the laws of nature. This then leads to the Poincare group (of special relativity) or the Einstein group (of general relativity), since these are Lie groups—groups of only continuous, analytical transformations of the spacetime coordinate systems that leave the laws of nature covariant. 

A C axis is said to be a proper rotation axis. An improper axis, S , exists when equivalency is restored by carrying out two operations consecutively a rotation about a proper axis, C , followed by reflection through a plane perpendicular to that axis. In the cyclobutane shown in Figure 2.10, for example, a C2 rotation about an axis perpendicular to the plane of the four carbon atoms, followed by a reflection through the plane containing the four carbon atoms, produces a structure identical to the starting structure. Note that neither the C2 rotation axis nor the o- reflection symmetry element is... 

As in low molecular weight compounds, optical activity can be observed only in chiral macromolecules, that is, macromolecules for which all allowed conformations lack reflection symmetry elements. 

A model of F is identical with its mirror image. It is achiral, although it does not have a plane of symmetry, due to the presence of a center of symmetry that is located between C2 and C3. A center of symmetry, like a plane of symmetry, is a reflection symmetry element. A center of symmetry involves reflection through a point a plane of symmetry requires reflection about a plane. A model of the mirror image of S (structure T) is not identical to S, but is a conformational enantiomer of S. They can be made identical 1 rotation about the C2, C3 bond. Since S and T are conformational enantiomers, the two will be present in equal amounts in a solution of this configurational stereoisomer. Both conformation S and conformation T are chiral and therefore should rotate the plane of plane polarized light. 

Inversion. Reflection through a point (Fig. 3.2). This point is the symmetry element and is called inversion center or center of symmetry. 

A geometric object can have several symmetry elements simultaneously. However, symmetry elements cannot be combined arbitrarily. For example, if there is only one reflection plane, it cannot be inclined to a symmetry axis (the axis has to be in the plane or perpendicular to it). Possible combinations of symmetry operations excluding translations are called point groups. This term expresses the fact that any allowed combination has one unique... 

An inversion center is mentioned only if it is the only symmetry element present. The symbol then is 1. In other cases the presence or absence of an inversion center can be recognized as follows it is present and only present if there is either an inversion axis with odd multiplicity (N, with N odd) or a rotation axis with even multiplicity and a reflection plane perpendicular to it (N/m, with N even). 

The mutual orientation of different symmetry elements is expressed by the sequence in which they are listed. The orientation refers to the coordinate system. If the symmetry axis of highest multiplicity is twofold, the sequence is x-y-z, i.e. the symmetry element in the x direction is mentioned first etc. the direction of reference for a reflection plane is nomal to the plane. If there is an axis with a higher multiplicity, it is mentioned first since it coincides by convention with the z axis, the sequence is different, namely z-x-d. The symmetry element oriented in the x direction occurs repeatedly because it is being multiplied by the higher multiplicity of the z axis the bisecting direction between x and its next symmetry-equivalent direction is the direction indicated by d. See the examples in Fig. 3.7. 

The occurrence of twinned crystals is a widespread phenomenon. They may consist of individuals that can be depicted macroscopically as in the case of the dovetail twins of gypsum, where the two components are mirror-inverted (Fig. 18.8). There may also be numerous alternating components which sometimes cause a streaky appearance of the crystals (polysynthetic twin). One of the twin components is converted to the other by some symmetry operation (twinning operation), for example by a reflection in the case of the dovetail twins. Another example is the Dauphine twins of quartz which are intercon-verted by a twofold rotation axis (Fig. 18.8). Threefold or fourfold axes can also occur as symmetry elements between the components the domains then have three or four orientations. The twinning operation is not a symmetry operation of the space group of the structure, but it must be compatible with the given structural facts. 

The nonvanishing components of the tensors y a >--eem and ya >-mee can be determined by applying the symmetry elements of the medium to the respective tensors. However, in order to do so, one must take into account that there is a fundamental difference between the electric field vector and the magnetic field vector. The first is a polar vector whereas the latter is an axial vector. A polar vector transforms as the position vector for all spatial transformations. On the other hand, an axial vector transforms as the position vector for rotations, but transforms opposite to the position vector for reflections and inversions.9 Hence, electric quantities and magnetic quantities transform similarly under rotations, but differently under reflections and inversions. As a consequence, the nonvanishing tensor components of x(2),eem and can be different... 

Lord Kelvin lla> recognized that the term asymmetry does not reflect the essential features, and he introduced the concept of chiralty. He defined a geometrical object as chiral, if it is not superimposable onto its mirror image by rigid motions (rotation and translation). Chirality requires the absence of symmetry elements of the second kind (a- and Sn-operations) lu>>. In the gaseous or liquid state an optically active compound has always chiral molecules, but the reverse is not necessarily true. 

For each symmetry element of the second kind (planes of reflection and improper axes of rotation) one counts according to Eq. (1) the pairs of distinguishable ligands at ligand sites which are superimposable by symmetry operations of the second kind. 

For information about point groups and symmetry elements, see Jaffd, H. H. Orchin, M. Symmetry in Chemistry Wiley New York, 1965 pp. 8-56. The following symmetry elements and their standard symbols will be used in this chapter An object has a twofold or threefold axis of symmetry (C2 or C3) if it can be superposed upon itself by a rotation through 180° or 120° it has a fourfold or sixfold alternating axis (S4 or Sh) if the superposition is achieved by a rotation through 90° or 60° followed by a reflection in a plane that is perpendicular to the axis of the rotation a point (center) of symmetry (i) is present if every line from a point of the object to the center when prolonged for an equal distance reaches an equivalent point the familiar symmetry plane is indicated by the symbol a. 

Figure 7.2 The different symmetry elements of the center ABg. (a) A trigonal axis, C3 (b) A binary axis, C2. (c) A symmetry axis belonging to both the 6C4 and SCj classes, (d) A symmetry plane, au- (e, f) Two of the six aj, reflection planes, (g) A view down the C3 axis in (a) to show a roto-reflection operation, S. ...

The interpretation of chemical reactivity in terms of molecular orbital symmetry. The central principle is that orbital symmetry is conserved in concerted reactions. An orbital must retain a certain symmetry element (for example, a reflection plane) during the course of a molecular reorganization in concerted reactions. It should be emphasized that orbital-symmetry rules (also referred to as Woodward-Hoffmann rules) apply only to concerted reactions. The rules are very useful in characterizing which types of reactions are likely to occur under thermal or photochemical conditions. Examples of reactions governed by orbital symmetry restrictions include cycloaddition reactions and pericyclic reactions. 

The symmetry elements will leave certain classes of reflections invariant, or F(H ) = F(H). Examination of Eq. (B.ll) shows that unless F(H) = 0, exp ( — 27nH-s) must be equal to 1, or... 

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