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Topic: Point Group c3, irreducible representation  (Read 671 times)

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nonscience_guy

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Point Group c3, irreducible representation
« on: February 13, 2020, 03:17:08 PM »
Hey!

I am a college student working on an app to help chemistry students get the solution to a point group. Similar to the webpage, http://gernot-katzers-spice-pages.com/character_tables/index.html .

I am currently having trouble get the solution for certain group points such as: c3, c4, c5, and c6.
When I go about inputting [24, 0] for c3 in my app it gives me the value of 8A + 16E, but the correct answer is 8A + 8E.

I use the given table in http://gernot-katzers-spice-pages.com/character_tables/C3.html , to calculate the answer. This is how I go about solving it.
=  coefficient * table value * input
A = 1 * 1 * 24 + 2 * 1 * 0 = 24 / h = 24 / 3 = 8 A
E = 1 * 2 * 24 + 2 * -1 * 0 = 48 /h = 48 / 3 = 16 E

I was wondering if anyone here could possibly show me a step by step solution on how to get the correct answer?

Corribus

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Re: Point Group c3, irreducible representation
« Reply #1 on: February 13, 2020, 04:55:44 PM »
Read the footnotes on the webpage you linked to. These groups are a little unusual:

"The single “E” re­presen­tation is reducible but almost behaves like a true irreducible re­presen­tation. Its norm, however, is twice the group order. Therefore, is has been marked with an asterisk in the table. This is essential when trying to decompose a reducible re­presen­tation into “irreducible” ones using the familiar projection formula."
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?  - Richard P. Feynman

nonscience_guy

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Re: Point Group c3, irreducible representation
« Reply #2 on: February 17, 2020, 05:29:10 PM »
Could you please elaborate a bit more?
Could you show a step by step solution using the projection formula?

Corribus

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Re: Point Group c3, irreducible representation
« Reply #3 on: February 18, 2020, 03:13:39 PM »
The two dimensional representation in the C3 character table is actually reducible, but you  must use complex numbers. Because complex numbers are generally undesirable in chemistry applications, the two dimensional representation is usually treated as basically irreducible and often expressed by combining the complex numbers with their complex conjugates to form a "real valued" representation.

The C3 character table is rigorously expressed as:

$$\begin{array}{c|ccc} C_4&\hat{E}&\hat{C_3}&\hat{C_3}^2 \\ \hline A&1&1&1\\ E&\lbrace\begin{array}{l}1\\1\end{array}&\begin{array}{l}\epsilon\\ \epsilon^*\end{array}&\begin{array}{l}\epsilon^*\\\epsilon\end{array}\rbrace \\ \end{array}$$

Where ε = exp(2πi/3).

Because these representations are actually reducible, you run into some issues when trying to use the various formulas and rules for manipulating character tables, including the reduction formula, that are intended for use with irreducible representations. You can go through and derive the results long hand, but if you replace h by 2h in the reduction formula it will give you the right answer when working with these pseudo-reducible 2-dimensional representations.
What men are poets who can speak of Jupiter if he were like a man, but if he is an immense spinning sphere of methane and ammonia must be silent?  - Richard P. Feynman