Symmetry operations, 7, 10 algebra

Sylow, 6. symbol list, xv. symmetric matrix, 59. symmetry element, 7. symmetry operations, 7, 10 algebra of, 15 for symmetric tripod, 27. symmetry operator, 10. symmetry orbitals, 203, 206, 207, 210, 212, 246. 

We now introduce the unity of the symmetry operator algebra as the symmetry operation which retains the status quo it is given the symbol Pf. 

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. 

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

Symmetry operations may be represented by algebraic equations. The position of a point (an atom of a molecule) in a Cartesian coordinate system is described by the vector r with the components x, y, z. A symmetry operation produces a new vector r with the components t, /, z. The algebraic expression representing a symmetry operation is a matrix. A symmetry operation is represented by matrix multiplication. 

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

Within the syntopy model, the essential algebraic structure of point symmetry groups is retained (in fact, this structure is extended), and the elements of syntopy groups are derived from ordinary point symmetry operators [252,394,395]. There are, however, alternative approaches for the generalization of symmetry, where fundamentally different algebraic structures are used. 

But this is eqivalent to the single operation <73. This can be represented as an algebraic relation among symmetry operators... 

Algebraic description of symmetry operations is based on the following simple notion. Consider a point in a three-dimensional coordinate system with any (not necessarily orthogonal) basis, which has coordinates x, y, z. This point can be conveniently represented by the coordinates of the end of the vector, which begins in the origin of the coordinates 0, 0, 0 and ends at x,y, z. Thus, one only needs to specify the coordinates of the end of this vector in order to fully characterize the location of the point. Any symmetrical transformation of the point, therefore, can be described by the change in either or both the orientation and the length of this vector. 

Therefore, symmetrical transformations in the crystal are formalized as algebraic (matrix-vector) operations - an extremely important feature used in all crystallographic calculations in computer software. The partial list of symmetry elements along with the corresponding augmented matrices that are used to represent symmetry operations included in each symmetry element is provided in Table 1.19 and Table 1.20. For a complete list, consult the Intemational Tables for Crystallography, vol. A. 

Table 1.20. Selected symmetry elements in trigonal and hexagonal crystal systems, their orientation and corresponding symmetry operations in the algebraic form as augmented matrices (see Figure 1.51). ...

We now consider how the two interacting symmetry operations produce a third symmetry operation, similar to how it was described in section 1.6 but now in terms of their algebraic representation. Assume that two symmetry operations, which are given by the two augmented matrices A and A, are applied in sequence to a point, coordinates of which are represented by the augmented vector V. Taking into account Eq. 1.48, but written in a short form, the first symmetry operation will result in the vector V given as... 

Second, algebraic differences between the equivalent positions for the space group are formed. For each pair of equivalent positions, one coordinate difference will turn out to be a constant, namely 0, 5, 3, 5, depending on the symmetry operator. These define the Harker sections for that space group, which are the planes having one coordinate u,v, or w constant, and that will contain peaks corresponding to vectors between symmetry equivalent atoms. In focusing attention only on Harker sections, the Patterson coordinates u,v,w... 

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