C2 character table

We have seen that any two of the C2, ( Jxz), (r Jyz) elements may be regarded as generating elements. There are four possible combinations of + 1 or — 1 characters with respect to these generating elements, + 1 and + 1, + 1 and -1,-1 and +1,-1 and —1, with respect to C2 and (tJxz). These combinations are entered in columns 3 and 4 of the C2 character table in Table A.l 1 in Appendix A. The character with respect to / must always be + 1 and, just as (r Jyz) is generated from C2 and (tJxz), the character with respect to (r Jyz) is the product of characters with respect to C2 and (tJxz). Each of the four rows of characters is called an irreducible representation of the group and, for convenience, each is represented by a symmetry species Aj, A2, or B2. The A] species is said to be totally symmetric since all the characters are + 1 the other three species are non-totally symmetric. 

The H2O molecule, therefore, has three normal vibrations, which are illustrated in Figure 4.15 in which the vectors attached to the nuclei indicate the directions and relative magnitudes of the motions. Using the C2 character table the wave functions ij/ for each can easily be assigned to symmetry species. The characters of the three vibrations under the operations C2 and (t (xz) are respectively + 1 and +1 for Vj, - - 1 and + 1 for V2, and —1 and —1 for V3. Therefore... 

Using the C2 character table (Table A. 11 in Appendix A) the characters of the vibrations under the various symmetry operations can be classified as follows ... 

Consider next the water molecule. As we have seen, it has a dipole moment, so we expect at least one IR-active mode. We have also seen that it has CIt, symmetry, and we may use this fact to help sort out the vibrational modes. Each normal mode of iibratbn wiff form a basis for an irreducible representation of the point group of the molecule.13 A vibration will be infrared active if its normal mode belongs to one of the irreducible representation corresponding to the x, y and z vectors. The C2 character table lists four irreducible representations A, Ait Bx, and B2. If we examine the three normal vibrational modes for HzO, we see that both the symmetrical stretch and the bending mode are symmetrical not only with respect to tbe C2 axis, but also with respect to the mirror planes (Fig. 3.21). They therefore have A, symmetry and since z transforms as A, they are fR active. The third mode is not symmetrical with respect to the C2 axis, nor is it symmetrical with respect to the ojxz) plane, so it has B2 symmetry. Because y transforms as Bt, this mode is also (R active. The three vibrations absorb at 3652 cm-1, 1545 cm-1, and 3756 cm-, respectively. 

We determined earlier (page 63) that the irreducible components of this representation arc three A, one /t2, two B,. and three B2 species. To obtain from this total set the representations for vibration only, it is necessary to subtract the representations for the other two forms of motion rotation and translation. We can identify them by referring to the C2 character table. The three translational modes will belong to the same representations as the x, y, and z basis functions, and the rotational modes will transform as R Rr, and R.. Subtraction gives... 

As environmental symmetry decreases, the orbitals will become split to an increasing extent. In the Cj point group, for example, all atomic orbitals become split into nondegenerate levels. This is not surprising since the C2, character table contains only one-dimensional irreducible representations. This result shows at once that there are no degenerate energy levels in this point group. This has been stressed in Chapter 4 in the discussion of irreducible representations. 

Thus, from the C2 character table (lower case letters are used for orbital representations, such that A a, B = b,. . )... 

Table Al.4.11 The character table of the molecular synnnetry group C2 (M)...

In the sixth column of the main body of the character table is indicated the symmetry species of translations (7) of the molecule along and rotations (R) about the cartesian axes. In Figure 4.14 vectors attached to the nuclei of H2O represent these motions which are assigned to symmetry species by their behaviour under the operations C2 and n (xz). Figure 4.14(a) shows that... 

Vibrations of the symmetry class Ai are totally symmetrical, that means all symmetry elements are conserved during the vibrational motion of the atoms. Vibrations of type B are anti-symmetrical with respect to the principal axis. The species of symmetry E are symmetrical with respect to the two in-plane molecular C2 axes and, therefore, two-fold degenerate. In consequence, the free molecule should have 11 observable vibrations. From the character table of the point group 04a the activity of the vibrations is as follows modes of Ai, E2, and 3 symmetry are Raman active, modes of B2 and El are infrared active, and Bi modes are inactive in the free molecule therefore, the number of observable vibrations is reduced to 10. 

The possible wave functions for the molecular orbitals for molecules are those constructed from the irreducible representations of the groups giving the symmetry of the molecule. These are readily found in the character table for the appropriate point group. For water, which has the point group C2 , the character table (see Table 5.4) shows that only A1 A2, B1 and B2 representations occur for a molecule having C2 symmetry. 

Follow the procedure used in the text in obtaining the character table for the C2 point group and develop the character table for the C3 point group. 

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

The A and B labels in Table 1 follow the convention that A s have characters of +1 for the rotation axis of highest order (C2 in the present case) while B s have character -1. A, by convention, is the totally symmetric i.r., since all operations of the group turn something of A symmetry into itself. Every group has a totally symmetric i.r. I.r. s with suffix 1 are symmetric (character +1) under av, whereas those with suffix 2 are antisymmetric (character -1). Table 1 is an example of a character table. Two-dimensional representations are denoted by symbols E. 

Consider the product

point group C2v, written as >1 ( 2) and that (p2 has symmetry B2, i.e. 4>2 B2). It is required to find the symmetry of tp =

wave function tp is subjected to all operations of C2v, starting with C2, noting from the character table that under this operation... 

One application of character tables is the identification of the symmetry species of given objects. For example, what is the symmetry species of a displacement Az in the positive z direction in the symmetry group C2v, assuming that the C2 axis is along z7... 

The symbols used for the representations are those proposed by Mulliken. The A representations are those which are symmetric with respect to the C2 operation, and the Bs are antisymmetric to that operation. The subscript 1 indicates that a representation is symmetric with respect to the ov operation, the subscript 2 indicating antisymmetry to it. No other indications are required, since the characters in the o column are decided by another rule of group theory. This rule is the product of any two columns of a character table must also be a column in that table. It may be seen that the product of the C2 characters and those of gv give the contents of the The representations deduced above must be described as irreducible representations This is because they... 

Individual molecular orbitals, which in symmetric systems may be expressed as symmetry-adapted combinations of atomic orbital basis functions, may be assigned to individual irreps. The many-electron wave function is an antisymmetrized product of these orbitals, and thus the assignment of the wave function to an irrep requires us to have defined mathematics for taking the product between two irreps, e.g., a 0 a" in the Q point group. These product relationships may be determined from so-called character tables found in standard textbooks on group theory. Tables B.l through B.5 list the product rules for the simple point groups G, C, C2, C2/, and C2 , respectively. 

The function x2 - y2 therefore has the following transformation properties under the symmetry elements of the group in the order E C2 1, -1, 1, 1, — 1. It is therefore assigned to the Bx representation. It is left as an exercise, which can be done in one s head, to assign the other five functions listed above. The results can be checked using the DA character table in Appendix II. 

Consider the group D3. Let the C3 axis coincide with the z axis and one of the C2 axes coincide with the x axis. Write out the complete matrices for all irreducible representations of this group. Derive from these the character table. 

It is easy to see that the operation C2 transforms into the negative of itself and v36 into itself. Thus a matrix is obtained which has only the diagonal elements -1 and 1 and the character 0 as required by the character table. It is equally easy to see that oh carries each component of i 3 into itself, so that the matrix of the transformation has only the diagonal elements I and 1 and hence a character of 2. We could carry out similar reasoning for the remaining operation applied to v, and v36 and also with respect to the application of all of the operations in the group to v and vAh, and it would be found that they satisfy the requirements of the characters of the E representation in every respect. 

The various symmetry operations will affect our set of C—O stretchings in the same way as they will affect the set of C—O bonds themselves. With this in mind we can determine the desired characters very quickly as follows. For the operation E the character equals 3, since each C—O bond is carried into itself. The same is true for the operation ah. For the operations C3 and S3 the characters are zero because all C—O bonds are shifted by these operations. The operations C2 and av have characters of 1, since each carries one C—O bond into itself but interchanges the other two. The set of characters, listed in the same order as are the symmetry operations at the top of the DVt character table, is thus as follows 3 0 1 3 0 1. The representation reduces to A + , as it should according to our previous discussion. Thus we have shown that normal modes of symmetry types A and E must involve some degree of C—O stretching. Since there is only one A mode, we can state further that this mode must involve entirely C—O stretching. 

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