Question

The molecule S2 is a bent molecule with C2y symmetry 1. С> (xz) S. aHowmany degrees of freedom are present in this molecule? bhHow many vibrational modes are possible? chComplete the character table below for the reducible representation of the bonds that remain unchanged in the molecule of SO2 for a C2v point group. Enter the total number of bonds that remain unchanged under the operation. Ifboth bonds change, enter a zero C2v C2 г dAier consulting a character table for the C2v point group, reduce the reducible representation to it irreducible representations ITY

Answer 1

a)

A molecule with N atoms has 3N degree of freedoms. Hence, since our molecule(SO2) has 3 atoms, the total number of degrees of freedom is 9.

b)

SO2 is a non-linear molecule. Hence, it has 3 translational and 3 rotational degrees of freedom. The remaining must be vibrational degrees of freedom.

Hence, the number of vibrational degrees of freedom in SO2 is

.

3-33

Hence, the answer is 3.

c)

E represents the identity operation. Hence, when we apply E on SO2, both the bonds remain unchanged. Hence, we put 2 in the table below E.

When we do a $C_2$, through z axis, i.e rotate the molecule by 180 degrees through z axis, we exchange the positions of the S-O bonds. Hence, both the bonds are changed. Hence, we put 0 under $C_2$.

When we put a reflection plane in the xz molecular plane, the plane cuts all the three atoms in half. Hence, the bonds stay unchanged. Hence, we put 2 under operation.

σ(πε)

When we put a reflection plane in the yz plane, it cuts the S atom in half and exchanges the position of the O atoms. Hence, the S-O bonds are exchanged in the process too. Hence, we put 0 under operation.

a(yz

Hence, the character table with the reducible representation is:

$C_{2v}$		$C_2$	σ(πε)	a(yz
$\Gamma$

d)

The character table of $C_{2v}$ point group is

$C_{2v}$		$C_2$	σ(πε)	a(yz
A	1	1	1	1
$A_2$	1	1	-1	-1
$B_1$	1	-1	1	-1
$B_2$	1	-1	-1	1

Now, the reducible representation we found in part c) is

$\Gamma : 2 \ 0 \ 2 \ 0$

Lets find out how many times the irreducible representation
A
occurs in our $\Gamma.$

The formula to find this is

$\frac{1}{h} \sum_{all\ classes} \chi_R \times \chi_I \times N$

Where h is the order of the group = 4 for C2v point group as it has 4 unique operations.

is the character of  the reducible representation, i.e 2, 0 , 2 and 0 for our reducible representations.

XR

$\chi_I$ is the character of the irreducible representation, i.e. for the first one, we are finding for , hence, the characters are 1, 1, 1 and 1 (look at the character table for the row A1.)

A

N is the number of symmetry operations in the particular class of operations.

hence, the number of times A1 occurs in our reducible representation is

$\frac{1}{h} \sum_{all\ classes} \chi_R \times \chi_I \times N \\ = \frac{1}{4}[2 \times 1 \times 1 + 0 \times 1 \times 1 + 2 \times 1 \times 1 + 0 \times 1\times 1 ] \\ \\ = \frac{1}{4} \times 4 = 1$

Hence, the irreducible rerpesentation A1 occurs only 1 time in our reducible representation.

Similarly, the number of times A2 occur in our reducible representation is

$\frac{1}{h} \sum_{all\ classes} \chi_R \times \chi_I \times N \\ = \frac{1}{4}[2 \times 1 \times 1 + 0 \times 1 \times 1 + 2 \times -1 \times 1 + 0 \times -1\times 1 ] \\ \\ = \frac{1}{4} \times [2+0-2+0] = 0$

Hence, A2 does not appear in our reducible representation.

The number of times B1 appear is

Σ XR x X N h all classes 1 12 x 1 x 10 x -1 x 1 +2 x 1 x 1 +0 x -1 x 1] x [20 2 0] = 1 Η

hence, B1 appears one time in the reducible representation.

The number of times B2 appear is

$\frac{1}{h} \sum_{all\ classes} \chi_R \times \chi_I \times N \\ = \frac{1}{4}[2 \times 1 \times 1 + 0 \times -1 \times 1 + 2 \times -1 \times 1 + 0 \times 1\times 1 ] \\ \\ = \frac{1}{4} \times [2+0-2+0] =0$

Hence, B2 does not appear in our reducible representation.

Hence,the reducible representation can be written as

$\Gamma = A_1 + B_1$

Note that the basis we took to calculate the reducible representation are the S-O bonds, Hence, the reducible representation actually represent the vibration of the S-O bonds in SO2.