Mathematical group symmetry operators

The sum of the diagonal elements of a matrix is called the character (%) of the matrix. Hereafter, we use the term character rather than the representation since there is a one-to-one correspondence between them and since mathematical manipulation with x is simpler than with the representation. The characters of the reducible representations for the E, and o operations are 3, 0 and 1, respectively. The characters for C3 (counterclockwise rotation by 120°) is the same as that of C, and those for cr2 and cr3 are the same as that of crj. By grouping symmetry operations of the same character ( class ), we obtain... 

It is seen that a group consisting of the mathematical elements (symmetry operations) /, A, B, C, D, and E satisfies the following conditions ... 

Once you can identify the specific symmetry operations a given molecule possesses you can then follow the flowchart to identify the proper name of the point group, here it is Cay. For this point group, there are four symmetry operations and we can state the mathematical definition of a point group while you visualize the operations in your mind. Chemists usually have developed minds-eye visualization due to studying organic chemistry and point group symmetry operations should be easy for chemists. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

The collection of all symmetry operations that leave a crystalline lattice invariant forms a space group. Each type of crystal lattice has its specific space group. The problem of enumerating and describing all possible space groups, both two dimensional three dimensional, is a pure mathematical problem. It was completely resolved in the mid-nineteenth century. A contemporary tabulation of the properties of all space groups can be found in Hahn (1987). Bums and Glazer (1990) wrote an introductory book to that colossal table. 

These six symmetry operations form a mathematical group. A group is defined as a set of objects satisfying four properties. 

The complete set of these operations constitutes a mathematical group, G, which, in three dimensions, is known as a space group. The symmetry operations... 

A mathematical group consists of a set of elements which are related to each other according to certain rules, outlined later in the chapter. The particular kind of elements which are relevant to the symmetries of molecules are symmetry elements. With each symmetry element there is an associated symmetry operation. The necessary rules are referred to where appropriate. 

The following four symmetry operations provide a set which will convert the molecule into itself and which form the mathematical group (C2v) of order four ... 

As already evident from the previous section, symmetry properties of a molecule are of utmost importance in understanding its chemical and physical behaviour in general, and spectroscopy and photochemistry in particular. The selection rules which govern the transition between the energy states of atoms and molecules can be established from considerations of the behaviour of atoms or molecules under certain symmetry operations. For each type of symmetry, there is a group of operations and, therefore, they can be treated by group theory, a branch of mathematics. 

It is apparent that we can always get a set of 3 x 3 matrices, which form a representation of a given point group, by consideration of the effect that the symmetry operations of the point group have on a position vector. Why this works is shown pictorially in Fig. 5-3.2 for the a = a TCt operation of, T. The symmetry operation C8 on the position veotor p followed by a r on p produces a vector p which is coincidental with the one produced by the operation o on p. The matrices D(Ct), D(a r), X>(o ) then simply mirror what is being done to the point vector. The general mathematical proof that, if symmetry operations R, S, and T obey the relation SR — T, then the matrices D(R), D(S), and D(T), found as above, obey the relation... 

Suppose that we have, by inspection, compiled a list of all of the symmetry elements possessed by a given molecule. We can then list all of the symmetry operations generated by each of these elements. Our first objective in this section is to demonstrate that such a complete list of symmetry operations satisfies the four criteria for a mathematical group. When this has been established, we shall then be free to use the theorems concerning the behavior of groups to assist in dealing with problems of molecular symmetry. 

Now, we can see that, because our set of operations is complete in the sense defined above, it satisfies the first requirement for mathematical groups, if we take as our law of combination of two symmetry operations the successive application of these operations. 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

It can be shown that a group consists of mathematical elements (symmetry elements or operations), and if the operation is taken to be performing one symmetry operation after another in succession, and the result of these operations is equivalent to a single symmetry operation in the set, then the set will be a mathematical group. The postulates for a complete set of elements A, B, C,... are as follows ... 

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