Groups and symmetry operators

We first define the notion of a group it is a finite or infinite set of operations, which satisfy the following four conditions  

Unit element there is one element E of the group for which EA = AE = A, where A is every other element of the group. 

Inverse element for every element A of the group, there exists another element called its inverse A for which AA = A A = E. 

A subgroup is a subset of operations of a group which form a group by themselves (they satisfy the above four conditions). Finally, two groups A and B are called isomorphic when there is a one-to-one correspondence between their elements Ai, A2. A and Bi, B2. B , such that 

A1A2 = A3 B1B2 = Bj For a crystal we can define the following groups  

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Behavior of Stereomodels under Permutations of Groups and Symmetry Operations, Pseudoasymmetry ... 

Modern methods in order to illustrate the nature of point groups and symmetry operations are available [16]. Molecules are abstracted as a set of points in space. We can associate with these points coordinates. 

Cell dimensions and standard uncertainties Space group and symmetry operators Reduced-cell parameters... 

SymApps converts 2D structures From the ChemWindow drawing program into 3D representations with the help of a modified MM2 force field (see Section 7.2). Besides basic visualization tools such as display styles, perspective views, and light source adjustments, the module additionally provides calculations of bond lengths, angles, etc, Moreover, point groups and character tables can be determined. Animations of spinning movements and symmetry operations can also he created and saved as movie files (. avi). 

In many extremely important cases, the analogy between a group of symmetry operations and a group of real numbers is more than superficial. For example, consider the molecule a—chloronaphthalene ... 

The members of symmetry groups are symmetry operations the combination rule is successive operation. The identity element is the operation of doing nothing at all. The group properties can be demonstrated by forming a multiplication table. Let us label the rows of the table by the first operation and the columns by the second operation. Note that this order is important because most groups are not commutative. The C3V group multiplication table is as follows ... 

For centrosymmetric systems with a centre of inversion /, subscripts g (symmetric) and u (antisymmetric) are also used to designate the behaviour with respect to the operation of inversion. The molecule trans-butadiene belongs to the point group Cik (Figure 2.13b). Under this point group the symmetry operations are /, C2Z, and i, and the following symmetry species can be generated ... 

What we have just done is to substitute the algebraic process of multiplying matrices for the geometric process of successively applying symmetry operations. The matrices multiply together in the same pattern as do the symmetry operations it is clear that they must, since they were constructed to do just that. It will be seen in the next section that this sort of relationship between a set of matrices and a group of symmetry operations has great importance and utility. 

The set ( bj) therefore closes. The other necessary group properties are readily proved and so G is a group. Direct product (DP) without further qualification implies the outer direct product. Notice that binary composition is defined for each group (e.g. A and B) individually, but that, in general, a multiplication rule between elements of different groups does not necessarily exist unless it is specifically stated to do so. However, if the elements of A and B obey the same multiplication rule (as would be true, for example, if they were both groups of symmetry operators) then the product at bj is defined. Suppose we try to take (a,-, bj) as a, bj. This imposes some additional restrictions on the DP, namely that... 

The set of components of the vector r1 in eq. (13) is the Jones symbol or Jones faithful representation of the symmetry operator R, and is usually written as (x / /) or x / z. For example, from eq. (15) the Jones symbol of the operator R (n/2 z) is (yxz) or yxz. In order to save space, particularly in tables, we will usually present Jones symbols without parentheses. A faithful representation is one which obeys the same multiplication table as the group elements (symmetry operators). 

A group-theoretical treatment of this symmetry contraint leads to the requirement that an MO must belong to an irreducible representation of the point group. A representation is a set of matrices - one for each symmetry operation - which constitutes a group isomorphous with the group of symmetry operations and can be used to represent the symmetry group. When we say that a function belongs to (or transforms as , or forms a basis for ) a particular representation, we mean that the matrices which constitute the representation act as operators which transform the function in the same way as the symmetry operations of the molecule. (The reader who knows little about matrices and their application as transformation operators can skip over such remarks.) An irreducible representation is one whose matrices cannot be simplified to sets of lower order. 

It has to be noted that the relation between the elements of 0(3)+ (also called SO(3), the group representing proper rotations in 3D coordinate space) and SU(2) (the special unitary group in two dimensions) is not a one-to-one correspondence. Rather, each R matches two matrices u. Molecular point groups including symmetry operations for spinors therefore exhibit two times as many elements as ordinary point groups and are dubbed double groups. 

If ra is an irreducible representation of dimension k, and if F, F2, is a set of degenerate eigenfunctions that form the basis for theyth irreducible representation of the group of symmetry operations, these eigenfunctions transform according to the relation... 

The Uniaxial or C. Gronps. These are the groups in which all operations are due to the presence of a proper axis as the sole symmetry element. The general symbol for such a group, and the operations in it, are... 

The group of symmetry operations form classes , is a class by itself A, B, C form a reflexion class D, F form a rotation class. Two elements P and Q satisfying the relation X PX=P or Q, where X is any element of the group and X its reciprocal, belong to the same class. This is seen to lead to the above results e.g. ... 

Suppose that G is the group of symmetry operations of a polyhedron or polygon, with vertices corresponding to the atomic positions in a particular molecular structure. The division of the structure into orbits, as sets of vertices equivalent under the actions of the group symmetry operations and the calculation of associated permutation representations/characters were described in Chapter 2. In this chapter, the identity between the permutation representa-tion/character on the labels of the vertices of an orbit and the a representation/character on sets of local s-orbitals or a-oriented local functions is exploited to constmct the characters of the representations that follow from the transformation properties of higher order local functions. 

Point group A group of symmetry operations that leave unmoved at least one point within the object to which they apply. Symmetry elements include simple rotation and rotatory-inversion axes the latter include the center of symmetry and the mirror plane. Since one point remains invariant, all rotation axes must pass through this point and all mirror planes must contain it. A point group is used to describe isolated objects, such as single molecules or real crystals. 

Symmetry Elements and Symmetry Operations 46 Point Groups and Molecular Symmetry 53 Irreducible Representations and Character Tables 59 Uses of Point Group Symmetry 63 Crystallography 74... 

One of the attractions of supramolecular chemistry is the extraordinary potential for synthesis of new materials that can be achieved much more rapidly and more effectively than with conventional covalent means. For supramolecular synthesis to advance, it is obviously important to characterize, classify, and analyze structural patterns, space group frequencies, and symmetry operators [118], However, at the same time we also need to bring together this information with the explicit aim of improving and developing supramolecular synthesis - the deliberate combination of different discrete molecular building blocks within periodic crystalline materials. 

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