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 isD3h 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)
471
The question asks for the frequency of the A'1 mode in the observed spectrum of BCl3, given its vibrational representation and the D3h character table.
First, let's understand the activity of vibrational modes in IR and Raman spectroscopy based on symmetry.
- A vibrational mode is IR active if it causes a change in the molecule's dipole moment. In terms of symmetry, this occurs if the mode transforms according to the same irreducible representation as the translational vectors (x, y, z). These are listed in the character table, often in the rightmost columns under the labels (x, y, z).
- A vibrational mode is Raman active if it causes a change in the molecule's polarizability ellipsoid. In terms of symmetry, this occurs if the mode transforms according to the same irreducible representation as the quadratic functions (x², y², z², xy, xz, yz, x² - y²). These are also listed in the character table, often in the rightmost columns under the labels like (x², y²), (z²), (xy), (xz, yz), (x² - y²).
The character table for the D3h point group is provided:
| 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 , xy) |
| 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 vibrational representation for BCl3 is given as $\Gamma_{vib} = A'_{1} + 2E' + A''_{2}$. Let's check the activity of each mode:
- A'1 mode: This mode transforms as (x² + y², z²). It is listed in the Raman activity column (quadratic functions). It is Raman active. It is not listed in the IR activity column (translational vectors x, y, z). It is IR inactive.
- E' modes (two sets): This mode transforms as (x, y) and (x² - y² , xy). It is listed in both the IR activity column ((x, y)) and the Raman activity column ((x² - y² , xy)). The E' modes are both IR active and Raman active.
- A''2 mode: This mode transforms as z. It is listed in the IR activity column (translational vector z). It is IR active. It is not listed in the Raman activity column (quadratic functions). It is Raman inactive.
Summarizing the activity of the fundamental modes:
- A'1: Raman active only
- E': IR active and Raman active
- A''2: IR active only
Now let's look at the observed spectra:
- Observed IR bands: 995, 480, and 244 cm-1
- Observed Raman bands: 995, 471, and 244 cm-1
Let's match the observed frequencies with the predicted activities:
- The frequency corresponding to the A'1 mode must be present in the Raman spectrum but absent in the IR spectrum (Raman active only). Comparing the lists, 471 cm-1 is present in the Raman spectrum but not in the IR spectrum.
- The frequency corresponding to the A''2 mode must be present in the IR spectrum but absent in the Raman spectrum (IR active only). Comparing the lists, 480 cm-1 is present in the IR spectrum but not in the Raman spectrum.
- The frequencies corresponding to the E' modes must be present in both the IR and Raman spectra (IR and Raman active). The frequencies 995 cm-1 and 244 cm-1 are present in both lists.
Based on this analysis, the frequency of the A'1 mode is 471 cm-1.
Detailed assignment:
- A'1 mode: 471 cm-1 (Raman active only)
- A''2 mode: 480 cm-1 (IR active only)
- E' modes: 995 cm-1 and 244 cm-1 (Both IR and Raman active)
Therefore, the frequency of the A'1 mode is 471 cm-1.
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 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,
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 :
