In the character table given below:
Td E $8C_3$ $3C_2$ $6S_4$ $6\sigma_d$ $A_1$ 1 1 1 1 1 $A_2$ 1 1 1 -1 -1 E 2 -1 2 0 0 $T_1$ 3 0 -1 1 -1 $T_2$ 3 0 -1 -1 1
The order of the point group is :
To determine the order of the point group \( T_d \), we need to sum the products of each column header (symmetry element) with its respective coefficient in the character table. Each symmetry element in a character table represents the number of symmetry operations of that type.
From the character table provided, we have the following symmetry elements and their coefficients:
- \(E\): Identity element, 1 operation.
- \(C_3\): 8 operations.
- \(C_2\): 3 operations.
- \(S_4\): 6 operations.
- \(\sigma_d\): 6 operations.
The order of a point group is calculated by summing these values:
\(Order = 1 + 8 + 3 + 6 + 6 = 24\)
Therefore, the order of the point group \( T_d \) is 24.
The correct answer is 24.
Let's briefly analyze why the other options are incorrect:
- 5: Too low for the complex symmetry of \( T_d \), which includes tetrahedral symmetry operations.
- 12: Insufficient to cover all operations including rotations and reflections in the tetrahedral group.
- 6: Represents basic mirror and rotation operations but misses combined ones like improper rotations.
Important Questions from Character Tables & Selection Rules
For the formaldehyde molecule, H2CO having C2v symmetry with the character table as given below,
C2v E C2 σv (xz) σv (yz) A1 1 1 1 1 z A2 1 1 -1 -1 Rz B1 1 -1 1 -1 x, Ry B2 1 -1 -1 1 y, Rx the reducible representation Γ3N (or Γtot) is Γ3N = 4A1 + A2 + 4B1 + 3B2. The reducible representation for the vibrational modes alone, namely Γvib will be
The reducible representation, Γ, in the table is equal to the following superposition of the irreducible representations of C2v point group.
C2v
E
C2
σv
\(\rm\sigma_{v}^{\prime}\) A1
1
1
1
1
A2
1
1
−1
−1
B1
1
−1
1
−1
B2
1
−1
−1
1
Γ
8
−2
−6
4
The observed IR spectrum for BCl3 exhibits three bands at 995, 480, and 244 cm-1, while the Raman bands are observed at 995, 471, and 244 cm-1. Given that for BCl3, Γvib = A'1 + 2E' + A2", the frequency of A1 mode in cm-1 is
D3h E 2C3 3C2 σh 2S3 3σv A'1 1 1 1 1 1 1 x2 + y2, z2 A' 2 1 1 -1 1 1 -1 Rz E' 2 -1 0 2 -1 0 (x, y) x2 - y2 , yz A" 1 1 1 1 -1 -1 -1 A" 2 1 1 -1 -1 -1 -1 z E" 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz) The character table for the point group D3h is given below.
D3h E 2C3 (z) \(\rm 3C_{2}^{'}\) σh(xy) 2S3 3σv \(\rm A_{1}^{'}\) +1 +1 +1 +1 +1 +1 - x2 + y2, z2 \(\rm A_{2}^{'}\) +1 +1 −1 +1 +1 −1 Rz - E' +2 −1 0 +2 −1 0 (x, y) (x2 − y2, xy) \(\rm A_{1}^{''}\) +1 +1 +1 −1 −1 −1 - - \(\rm A_{2}^{''}\) +1 +1 −1 −1 −1 +1 z - E'' +2 −1 0 −2 +1 0 (Rx, Ry) (xz, yz) In the electronic ground state, BF3 has D3h symmetry. Therefore,
