The following character table (accompanied by a molecule of the corresponding point group) has some missing information. 2 C3 3 C2 2 S3 3 ov X1 1 1 1 1 1 1 x2+ y?, z2 1 1 -1 1 1 -1 Rz E' 2 -1 2 -1 (x, y) (x2- y?, xy) A;" 1 X2 1 -1 -1 -1 Az" 1 1 -1 -1 -1 1 X3 BH; E" -1 -2 1 (Rx, Ry) (2х, yz) а. Based on symmetry labels and Mulliken symbols, what is the missing symmetry element? b. Provide the missing information in the table: i. X1 = ii. X2 = iii. X3 = С. For this point group, what is the symmetry of the following orbitals for an atom located along the main C3 axis? i. A px orbital: ii. A d? orbital: iii. A dxy orbital:
The following character table (accompanied by a molecule of the corresponding point group) has some missing information. 2 C3 3 C2 2 S3 3 ov X1 1 1 1 1 1 1 x2+ y?, z2 1 1 -1 1 1 -1 Rz E' 2 -1 2 -1 (x, y) (x2- y?, xy) A;" 1 X2 1 -1 -1 -1 Az" 1 1 -1 -1 -1 1 X3 BH; E" -1 -2 1 (Rx, Ry) (2х, yz) а. Based on symmetry labels and Mulliken symbols, what is the missing symmetry element? b. Provide the missing information in the table: i. X1 = ii. X2 = iii. X3 = С. For this point group, what is the symmetry of the following orbitals for an atom located along the main C3 axis? i. A px orbital: ii. A d? orbital: iii. A dxy orbital:
Chemistry
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Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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Transcribed Image Text:## Character Table Analysis for Point Group Symmetry
### Missing Information in Character Table
The table below is a character table for a point group associated with a molecule (BH₃). Some information, including symmetry elements and functions, is missing and needs to be determined.
| | E | 2 C₃ | 3 C₂ | ? | 2 S₃ | 3 σᵥ | Basis Functions |
|------|---|------|------|---|-----|------|-----------------|
| X₁ | 1 | 1 | 1 | 1 | 1 | 1 | x² + y², z² |
| A₂' | 1 | 1 | -1 | 1 | 1 | -1 | Rz |
| E’ | 2 | -1 | 0 | 2 | -1 | 0 | (x, y) |
| A₁’’ | 1 | X₂ | 1 | -1 | -1 | -1 | |
| A₂’’ | 1 | 1 | -1 | -1 | -1 | 1 | X₃ |
| E’’ | 2 | -1 | 0 | -2 | 1 | 0 | (Rx, Ry) |
- **Symmetry Operations:**
- **C₃**: 120-degree rotation about the principal axis.
- **C₂**: 180-degree rotation about a secondary axis.
- **σᵥ**: Vertical mirror plane.
- **S₃**: Improper rotation.
### Questions
**a. Missing Symmetry Element:**
Identify the missing symmetry element represented by "?".
**b. Missing Information in Table:**
Determine the unknown values for X₁, X₂, and X₃.
- **i. X₁ =**
- **ii. X₂ =**
- **iii. X₃ =**
**c. Orbital Symmetry Along C₃ Axis:**
For this point group, identify the symmetry of the following orbitals for an atom located along the main C₃ axis:
- **i. A pₓ orbital: ___________**
- **ii. A d
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