What is the Direct product of Eg x Eu X A1g where the order is final, transition operator, and initial respectively? We are looking at a molecule that is in the D3d group
What is the Direct product of Eg x Eu X A1g where the order is final, transition operator, and initial respectively? We are looking at a molecule that is in the D3d group
Chapter27: Molecular Fluorescence Spectroscopy
Section: Chapter Questions
Problem 27.3QAP
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Transcribed Image Text:What is the Direct product of Eg x Eu X A1g
where the order is final, transition operator,
and initial respectively? We are looking at a
molecule that is in the D3d group
Expert Solution
Step 1
In group theory, a branch of abstract algebra, a character table is a two-dimensional table whose rows correspond to irreducible representations, and whose columns correspond to conjugacy classes of group elements.
If are 2 irreducible representation of a group , then the direct product representation is obtained by multiplication of the characters of 2 representation under each symmetric operation. In direct product only characters are to be multiplied not the order of the class.
Step 2
The character table of is :
| D3d | E | 2C3 | 3C'2 | i | 2S6 | |
| A1g | +1 | +1 | +1 | +1 | +1 | +1 |
| A2g | +1 | +1 | -1 | +1 | +1 | -1 |
| Eg | +2 | -1 | 0 | +2 | -1 | 0 |
| A1u | +1 | +1 | +1 | -1 | -1 | -1 |
| A2u | +1 | +1 | -1 | -1 | -1 | +1 |
| Eu | +2 | -1 | 0 | -2 | +1 | 0 |
The direct product of Eg * Eu * A1g is:
| Eg | +2 | -1 | 0 | +2 | -1 | 0 |
| Eu | +2 | -1 | 0 | -2 | +1 | 0 |
| A1g | +1 | +1 | +1 | +1 | +1 | +1 |
| +4 | +1 | 0 | -4 | -1 | 0 |
Since it is reducible representation , it need to be converted into irreducible representation.
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