The point group of borane (BH3) is D3h. The reducible representation for the motions of BH3 and character table for D3h are given below. Find the irreducible representations that make up the reducible representation
The Correct Answer and Explanation is :
o determine the irreducible representations that compose the reducible representation for the motions of borane (BH₃), we utilize the character table for the D₃h point group.he character table provides the characters for each irreducible representation under various symmetry operations.
Character Table for D₃h Point Group:
| D₃h | E | 2C₃ | 3C₂’ | σₕ | 2S₃ | 3σᵥ | Functions |
|---|---|---|---|---|---|---|---|
| A₁’ | 1 | 1 | 1 | 1 | 1 | 1 | z², x²+y² |
| A₂’ | 1 | 1 | -1 | 1 | 1 | -1 | Rz |
| E’ | 2 | -1 | 0 | 2 | -1 | 0 | (x, y) |
| A₁” | 1 | 1 | 1 | -1 | -1 | -1 | |
| A₂” | 1 | 1 | -1 | -1 | -1 | 1 | z |
| E” | 2 | -1 | 0 | -2 | 1 | 0 | (Rx, Ry) |
| iven the reducible representation Γ for the motions of BH₃, we aim to express Γ as a sum of the irreducible representations (irreps) of the D₃h point group.his process is known as reducing the reducible representation. | |||||||
| Steps to Reduce the Reducible Representation: |
- Identify the Reducible Representation (Γ):
he reducible representation Γ corresponds to the 9 degrees of freedom in BH₃ (3 atoms × 3 degrees of freedom per atom). - Calculate the Characters for Each Symmetry Operation:
or each symmetry operation in the D₃h point group, determine how many atoms remain unmoved and how their coordinates transform.um the contributions to obtain the character for each operation. - Apply the Reduction Formula:
he number of times an irreducible representation (χᵢ) appears in the reducible representation (Γ) is given by:
[ a_i = \frac{1}{h} \sum [n_k \cdot \chi_i(R_k) \cdot \Gamma(R_k)] ]
Where:
- ( h ) is the order of the group (for D₃h, ( h = 12 )). – ( n_k ) is the number of operations in the ( k )-th class. – ( \chi_i(R_k) ) is the character of the irreducible representation for the ( k )-th operation. – ( \Gamma(R_k) ) is the character of the reducible representation for the ( k )-th operation.
- Compute the Coefficients (aᵢ):
erform the summation for each irreducible representation to find its coefficient in the reducible representation.
Example Calculation:
et’s calculate the coefficient for the A₁’ irreducible representation:
- Characters for A₁’: 1, 1, 1, 1, 1, 1]- Characters for Γ: 9, 0, -1, 3, 0, 1] (hypothetical values for illustration)- Number of operations in each class: 1, 2, 3, 1, 2, 3]
[ a_{A_1′} = \frac{1}{12} \left(1 \cdot 1 \cdot 9 + 2 \cdot 1 \cdot 0 + 3 \cdot 1 \cdot (-1) + 1 \cdot 1 \cdot 3 + 2 \cdot 1 \cdot 0 + 3 \cdot 1 \cdot 1\right) ]
[ a_{A_1′} = \frac{1}{12} (9 + 0 – 3 + 3 + 0 + 3) = \frac{1}{12} \times 12 = 1 ]
his indicates that the A₁’ irreducible representation appears once in the reducible representation Γ.
y performing similar calculations for the other irreducible representations, we can determine the complete decomposition of Γ.
Conclusion:
he reducible representation Γ for the motions of BH₃ can be decomposed into the irreducible representations of the D₃h point group.his decomposition reveals the symmetry properties of the molecular vibrations and is essential for understanding the vibrational spectroscopy of the molecule.