Improper rotations definition

This fact explanation resides in the improper rotation definition, which for file even orders (2 , = 3 for the current case) generates the -Cll xcr " -E actually regaining the initial state after... 

It will be shown later that the operations of rotation and reflection in a plane perpendicular to the rotation axis always give the same result regardless of the order in which they are performed. Thus the definition of improper rotation need not specify the order. 

The rigorous group theoretical requirement for the existence of chirality in a crystal or a molecule is that no improper rotation elements be present. This definition is often trivialized to require the absence of either a reflection plane or a center of inversion in an object, but these two operations are actually the two simplest improper rotation symmetry elements. It is important to note that a chiral object need not be totally devoid of symmetry (i.e., be asymmetric), but that it merely be diss)nn-metric (i.e., containing no improper rotation symmetry elements). The tetrahedral carbon atom bound to four different substituents may be asymmetric, but the reason it represents a site of chirality is by virtue of dissymmetry. 

One definition of chirality is that the molecule be nonsuperimposablc on its mirror image. An equivalent criterion is that it not possess an improper axis of rotation (page 52). The absence of a mirror plane does not insure optical activity because a molecule may have no mirror plane, yet may possess an improper rotational axis. We can, hpv/ever, be sure that the molecule with a mirror plane will be optically inactive. 

By adjoining a definite translation to a point symmetry operation (that is, a proper or improper rotation) we obtain a space transformation. If a system is sent into itself by such a combination of operations the com-... 

We suppose that jR is a symmetry operator that corresponds to some proper or improper rotation, and that r is a vector in the real lattice. The vector Rr is also a vector in the real lattice since K is a symmetry operator. There are as many points in reciprocal lattice space as in the direct lattice, and each direct lattice vector corresponds to a definite vector in the reciprocal lattice. It follows that Rr corresponds to a reciprocal lattice point if r is a reciprocal lattice vector. Thus the operators R, S,. . . , that form the rotational parts of a space group are also the rotational parts of the reciprocal lattice space group. It now follows that the direct and reciprocal lattices must belong to the same crystal class, although not necessarily to the same type of translational lattice (see Eqs. 10.28-10.31). 

The space group of a crystal structure can be considered as the set of all the symmetry operations which leave the structure invariant. All the elements (symmetry operations) of this set satisfy the characteristics of a group and their number (order) is infinite. Of course, this definition is only valid for an ideai structure extending to infinity. For practical purpose, however, it can be applied to the finite size of real crystals. Lattice translations, proper or improper rotations with or without screw or gliding components are all examples of symmetry operations. 

Applying two improper rotations to a molecule yields the same molecule , so we can speak of the mirror image as the result of reflection at any plane. The reason is that two improper rotations constitute a proper rotation, a linear mapping of determinant +1, and, by definition, we consider molecules in space as equal if they arise from each other by a translation plus a proper rotation. Thus, chirality can be considered as an... 

SOLUTION For operators involving reflection through one of the Cartesian planes or Q rotation about a Cartesian axis, it is often simplest to write what is happening in terms of the Cartesian coordinates. In this example, if we assign the proper rotation axis to the z axis, then the horizontal mirror plane (defined to be perpendicular to the principal axis) is the xy plane. Then from Eqs. 6.1 and 6.3 we have C2 (x,y,x) = tjj(—x,—y,z) and d- ,i/>(x,y,z) = ifi(x,y,—z). Combining these with the definition of the improper rotation, we find... 

There are several examples of simple C2 symmetric molecules, including hydrogen peroxide (H2O2, Figure 3.8a) and (15, 25 )-l,2-dimethylcyclopropane (Figure 3.8b). By definition, C groups do not contain improper rotation axes, and so molecules in these... 

B As we have. seen. Hie formal definition of optical activity is bused upon the absence of an improper axis of rotation. The Iwo definitions arc equivalent. 

As we have seen, the formal definition or optical activity is based upon the absence off an improper axis of rotation. The two definitions are equivalent. 

Chemists are more used to the operational definition of symmetry, which crystaUo-graphers have been using long before the advent of quantum chemistry. Their ball-and-stick models of molecules naturally exhibit the symmetry properties of macroscopic objects they pass into congruent forms upon application of bodily rotations about proper and improper axes of symmetry. Needless to say, the practitioner of quantum chemistry and molecular modeling is not concerned with balls and sticks, but with subatomic particles, nuclei, and electrons. It is hard to see how bodily rotations, which leave all interparticle distances unaltered, could affect in any way the study of molecular phenomena that only depend on these internal distances. Hence, the purpose of the book will be to come to terms with the subtle metaphors that relate our macroscopic intuitive ideas about symmetry to the molecular world. In the end the reader should have acquired the skills to make use of the mathematical tools of group theory for whatever chemical problems he/she will be confronted with in the course of his or her own research. 

This is also called a rotai j -reflcction axis, or an improper axis. In some applications, notably E. Wignor, Ges. IFfss. Gottingen, p. 133 (1930), a different definition is used in yhieh the molecule is first rotated and then inverted through a center, rather than reflected in a plane. The (wo definitions are closely related. 

A more rigorous symmetry-based definition is that a chiral molecule does not have an improper axis of rotation (S ). There are several ligands... 

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