Point groups identity operation

If a basis of functions is chosen on the reasonable grounds that symmetry-equivalent atoms in the molecule have identical basis functions centred on them, then this basis will carry a representation of the molecular point group any operation of the point group Q ) will send the basis functions into linear combinations of themselves without the generation of any functions outside the basis. 

Invariance under the point-group operations requires that the crystal-field Hamiltonian contain only operators that transform as the identity representation of the point group. These operators are easy to determine in general, since, for all the point groups except the cubic groups (T, Tj, T, O, and Oh), all group operators may be constructed from the following operators (Leavitt, 1980) ... 

In this D2h case, unlike in the C- y point group, each operation is in its own class, and the number of columns above is identical to that in the >2 character table. SALC(Ag) and SALC(5i ) of these oxygen 2s wave functions can be obtained by multiplication of each outcome by the characters associated with each operation of these irreducible representations, followed by addition of the results ... 

Each bond orbital in a molecule may be described by a function that is localized in the bond region. If the bond is of rotations about the bond axis, and reflections in planes containing the bond axis. The x bond functions are antisymmetric to 180 rotations about the bond axis, and antisymmetric to reflections in the molecular plane or in the bond plane. A set of equivalent orbitals of this type give a reducible representation of the molecxilar point group since operations that interchange identical nuclei also interchange equivalent bond functions. Some aspects of this procedure were already outlined, leading to Eq. 7.119. 

The structural characteristics of the polymer chain constructed as described above are that translational or roto-translational symmetry can be found along the chain in addition to the traditional point group symmetry operations (rotation axes, symmetry planes, inversion center, and identity) [18]. 

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. 

If o is chosen as the generating function, it yields the other two members of the set (as well as itself) under the symmetry operations of the point group. The function o is obviously the result of the identity operation, while Cj and Cy produce 02 and <73, respectively. These three symmetry operations are in fact sufficient to resolve the problem, although the reader can verify that if all of file operations of the group are employed, the same expression will be obtained (problem 17). 

This is the highest multiplicity Mmax of the given space group and corresponds to the lowest site symmetry (each point is mapped onto itself only by the identity operation ). In this general position the coordinate triplets of the Mmax sites include the reference triplet indicated as x, y, z (having three variable parameters, to be experimentally determined). In a given space group, moreover, it is possible to have several special positions. In this case, points (atoms) are considered which... 

The six operations for the symmetric tripod form a particular point group and the way in which they combine together is conveniently summarized by what is known as a group table (Table 3-4.1). In this table the operation to be first carried out is given in the first row and the second operation to be carried out in the first column, the combination falls in the body of the table at the intersection of the appropriate row and column, e.g. o o = C8, that is the operation o followed by the operation a T is identical to the single operation C8 (see Fig. 3-4.2). 

In Table 6-3.1 we show the matrices for all of the operations of the 8v point group using both real and complex p-orbitals as basis functions. For the operations Ct and Cj we have simply replaced 0 by 27 /3 and 4t /3 respectively in both eqn (6-3.1) and eqn (6-3.2). The matrices for the rejection operations have been obtained in a fashion similar to that used for the rotations. In carrying out these steps it has been assumed that plf p, and p lie along the vectors 6t, e8, and e, respectively (see Fig. 6-3.1). For obvious reasons the matrix representation in the real basis is identical to the one given in 5-3(2) and, further, the reader may verify for himself that the matrices using the complex basis obey the 8v group table (Table 3-4.1). 

Since we will continually be requiring the characters of the irreducible representations of the point groups, it is convenient to put them together in tables known as character tables- In the character table of a point group each row refers to a particular irreducible representation and, since the characters of operations of the same class are identical, only a single entry (C,) is made for all the operations of a given class. The columns are headed by a representative element from each class preceded by the number of elements or operations in that class gf. 

If the point group has a oh operation but no i operation (groups nb and B with n odd) the labels are primed or double primed according to whether the character of oh is positive or negative, respectively. The situation is similar to the one in the previous paragraph in that oh will be represented by +1 or -1 times an identity matrix and that there are twice as many irreducible representations in B as in [Pg.132]

We start with the 3N unit displacement vectors, where there are three mutually perpendicular vectors on each nucleus. Consider the effect of a symmetry operation on one of these 3N vectors. Each vector either will remain on the same nucleus, with or without its direction changed, or it will be moved to an identical nucleus, with or without its direction changed. (We consider the nuclei as fixed while examining the effect of a symmetry operation on a displacement vector see Section 6.3.) Since any vector on a nucleus can be expressed as some linear combination of the three mutually perpendicular unit displacement vectors on that nucleus, a symmetry operation will send each of the 3N displacement vectors into some linear combination of these 3 N vectors. Therefore, the unit displacement vectors form a basis for some SW-dimensional representation of the molecular point group we shall call this representation r3Jv. 

Second, a multiplication table for the factor group is written down. The space group formed by the above symmetry elements is infinite, because of the translations. If we define the translations, which carry a point in one unit cell into the corresponding point in another unit cell, as equivalent to the identity operation, then the remaining symmetry elements form a group known as the factor, or unit cell, group. The factor... 

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