C2h character table

Let us check these rules on the C2h character table given above. All four irreducible representations have 1 as their character under E, so all of them are one-dimensional. Applying rule 1,... 

Both representations we constructed here are reducible since there are no 2- and 12-dimensional representations in the C2h character table (Table 4-7). The next question is how to reduce these representations. 

These 12 irreducible representations account for the 12 degrees of motional freedom of HNNH. Subtracting the irreducible representations corresponding to the translation and rotation of the molecule (see C2h character table, Table 5-2) leaves us the symmetry species of the normal modes of vibration ... 

Table 5-2. The C2h Character Table and the Representations of the Internal Coordinates of Diimide...

To find the irreproducible representations that these orbitals span, we multiply the characters of orbitals by the characters of the irreproducible representations, sum those products, and divide the sum by the order h of the group (as in Section 12.5(a)). The table below illustrates the procedure, beginning at left with the C2h character table. 

FIGURE 13. Character tables for symmetry point groups C2h, C2 and C2v... 

Table 4-3. A Preliminary Character Table for the C2h Point Group...

The character tables usually consist of four main areas (sometimes three if the last two are merged), as is seen in Table 4-5 for the C3v and in Table 4-7 for the C2h point group. The first area contains the symbol of the group (in the upper left corner) and the Mulliken symbols referring to the dimensionality of the representations and their relationship to various symmetry operations. The second area contains the classes of symmetry operations (in the upper row) and the characters of the irreducible representations of the group. 

Trans-N2p2 has C2h symmetry and its coordinates are taken as those shown below. The character table of the C2h point group is shown below. 

This is followed by a list of all symmetry types possible for the vibrational modes. This is simply a list of the irreducible representations present in the character table for the molecule s symmetry point group for trans-l,2-dichloroethane, having C2h symmetry, this is Ug, bg (symmetric motions) and o , ba (antisymmetric). 

Two structures have been proposed for (Gly) I an antiparallel-chain pleated sheet (APPS) and a similar rippled sheet (APRS) (see Section III,B,1). These structures have different symmetries the APPS, with D2 symmetry, has twofold screw axes parallel to the a axis [C (a)] and the b axis [C (b)], and a twofold rotation axis parallel to the c axis [62(0)] the APRS, with C2h symmetry, has a twofold screw axis parallel to the b axis ( 2(6)], an inversion center, i, and a glide plane parallel to the ac plane, o-Sj. Once these symmetry elements are known, together with the number of atoms in the repeat, it is possible to determine a number of characteristics of the normal modes the symmetry classes, or species, to which they belong, depending on their behavior (character) with respect to the symmetry operations the numbers of normal modes in each symmetry species, both internal and lattice vibrations their IR and Raman activity and their dichroism in the IR. These are given in Table VII for both structures. 

Overtone and combination bands belong to symmetry species determined by the species of their fundamentals. We can determine this symmetry by multiplying the characters for the fundamentals. Thus, overtones of all species have the character of the totally symmetric species, A for D2 symmetry and Ag for 21, symmetry (see Table VII). For combinations, this rule implies that, for >2 symmetry, a Bi mode combining with a Bs mode produces a combination of B2 symmetry, etc., while for C2h, Bu combining with Bg results in a band of Au symmetry, etc. (see Table VII). 

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