C4 character table

Using the C4, character table, determine the symmetry labels (labels of irreducible representations) of these orbitals. 

Let us consider the character table of the C4 group and the representation F displayed below (see Table 7.3). F is, reducible, because its character does not coincide with any one of the irreducible representations of the Cav group. According to Equation (7.4) and the character table (Table 7.3), we can write... 

Table 7.3 The character table of group C4 . The basis functions are not included for the sake of brevity. A reducible representation, F, is shown below...

Work out the character table for the group C4. Present it in the conventional format, reducing all exponentials to their simplest form. 

With these considerations in mind, the process of constructing the correct linear combinations of the subsets proceeds exactly as in the case of the carbocyclic systems. The correct coefficients of the atomic orbitals are simply the characters of the representations. For the E orbitals we will obtain some imaginary coefficients, but these may be eliminated by taking the appropriate linear combinations. We can thus write, almost by direct inspection of the character table of the C4 group ... 

Exercise 13.4-7 Determine if time-reversal symmetry introduces any additional degeneracies in systems with symmetry (1) C3 and (2) C4, for (i) Neven and (ii) Nodd. [Hints Do not make use of tabulated PFs but calculate any PFs not already given in the examples in Section 12.4. Characters may be found in the character tables in Appendix A3.]... 

Given the illustrations of the normal modes of a molecule, it is possible to identify their symmetry species from the character table. Each non-degenerate normal mode can be regarded as a basis, and the effects of all symmetry operations of the molecular point group on it are to be considered. For instance, the v mode of XeF4 is invariant to all symmetry operations, i.e. R(v 1) = (l)vj for all values of R. Since the characters are all equal to 1, v belongs to symmetry species Aig. For V2, the symmetry operation C4 leads to a character of -1, as shown below ... 

If we consider the two n antibonding combinations of the lone pairs, we notice that they are related by a rotation of 90" about the z-axis, which is a symmetry element of the system (a C4-axis). The same applies for the two tt-bonding combinations, and also for the xz and yz orbitals on the metal. These are pairs of orbitals that are degenerate by symmetry in the D4j, point group. If we consult the character table for... 

The CIOFJ anion has a square pyramidal geometry, belonging to the point group C4, with the character table shown. The ground state MO configuration is [ core where the core has no unfilled orbitals, and the lowest excited state MO configuration is [ core ] e Ui. ... 

The entries 2C4, and 2aj in the top row of the character table for the C4V point group mean that this point group has two independent symmetry operations in each of the C4, a and a classes. The C4 class includes both the C4 operation itself, and the inverse of this operation, C4 , which is the same as 4. The basis functions for the two-dimensional irreducible representation (E) in the last row are pairs of coordinate values (x, y) or pairs of products of these values. The character 2 here means that the identity symmehy preserves both values, as it should, and the character -2 indicates that the C2 operation changes the sign of both values. 

Figure 3.7 The top lines of the character tables for (a) C4 and (b) S4 point groups.

Table 4.8 shows the standard character table for the I)4h point group. The top row of the character table is the list of the operations valid in this point group. This list always begins with the identity operator, and in groups like O41, which contains a rotational subgroup, the rotational operations are given next. The principal axis is the C4 axis and the corresponding... 

Although the C4 and 4 matrices have different off-diagonal elements, the trace of each is 1. In fact, the trace of the matrices for these operations would be equal irrespective of the basis chosen. A character table only lists the characters for the standard sets of representations, and so the C4 and 4 operations can be put together in a single column. In this way the columns of the character tables may contain more than one operation and the operations contained in any column are linked by having the same character. The columns of the charaeter table contain classes of operations which may be sets of one or more actual operations. This example shows that, in >4, a 90° rotation clockwise (C4 )... 

The first type has operations that are linked to the same symmetry element, such as C3 and C3. However, operations linked to the same element will not always fall into the same class for example, in >41, the C4 and rotations associated with the principal axis are in the same class, but the 4 operation is listed separately in the character table as C2. The second types of operation that fall into the same class are those for sets of symmetry-equivalent elements, such as the three equivalent mirror planes in >31,. 

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