C4v character table

This representation can be reduced to rows of characters from the C4V character table ... 

The vibrational modes of M(CO)5X therefore have Ai, Bj and E symmetries and the C4V character table shows that only the two Ai and the E modes are IR active, consistent with the observation of three absorptions in the IR spectrum. 

At the point A along EX, P(k) is C4v which is isomorphous with D4. The character table (Appendix A3) is therefore the same as the upper left quadrant of D4h but the basis... 

Exercise 17.4-5 AtX, P(k) is C2v the character table for which is shown in Table 17.7. The basis functions shown are those for the IRs of C2v when the principal axis is along a. Table 17.8 contains the character table for C3V with basis functions for a choice of principal axis along [1 1 1], The easiest way to transform functions is to perform the substitutions shown by the Jones symbols in these two tables. The states X2 and S4 are antisymmetric with respect to Jones symbol for ah in Table 17.7). Note that vertical planes at A and, as Table 17.8 shows, A2 is antisymmetric with respect to dihedral planes in C4v and from Table 17.2 we see that the bases for T j and T2 are antisymmetric with respect to fold axis at T is parallel to kz, and carrying out the permutation y —> z, z —> x, x y on the bases for A[ and A2 gives xy(x2 y2) and x2 y2 for the bases of Tj and T2, which are antisymmetric with respect to ab.)... 

The additional symmetry elements in the silicon structure are given in Table 17.12. At A, P(k) C4V, which is isomorphous with Gg. Jones symbols and the character table are in Table 17.15. 

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. 

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