**Problem 1:****Objective:** Using the provided character table for the point group \(D_{3h}\), determine the irreducible representations contained in the reducible representation \(\Gamma\). Show your work.---**Character Table for \(D_{3h}\) (6m2):**| Representation | \(E\) | \(2C_3\) | \(3C_2\) | \(\sigma_h\) | \(2S_3\) | \(3\sigma_v\) | Basis Functions ||----------------|-------|---------|---------|--------------|---------|--------------|----------------|| \(A_1'\) | 1 | 1 | 1 | 1 | 1 | 1 | \(x^2+y^2, z^2\) || \(A_2'\) | 1 | 1 | -1 | 1 | -1 | -1 | \(R_z\) || \(E'\) | 2 | -1 | 0 | 2 | -1 | 0 | \((x, y)\) || \(A_1''\) | 1 | 1 | 1 | -1 | -1 | -1 | || \(A_2''\) | 1 | 1 | -1 | -1 | 1 | 1 | \(z\) || \(E''\) | 2 | -1 | 0 | -2 | 1 | 0 | \((R_x, R_y)\), \((zx, yz)\) |**Given:** \(h = 12\)---**Reducible Representation \(\Gamma\):**\[\Gamma = [6, 3, -2, 2, -1, -2]\]

Answered: Image /qna-images/answer/ef28280a-4031-471e-bc1b-5dd295a20337.jpg

1. Provided the following character table for the point group D3h, the irreducible representations contained determine in the reducible representation r. Show your work. D3h (6m2) E 2C, 3C, 25, 30, h = 12 A x² + y°, z? 1 1 1 1 1 1 A 1 1 -1 1 1 -1 R. E' -1 -1 (x, y) (x² – y², xy) A" A" 1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 E" -1 -2 1 (R,, R,) (zx, yz) 6 3 -2 2 -1 -2

1. Provided the following character table for the point group D3h, the irreducible representations contained determine in the reducible representation r. Show your work. D3h (6m2) E 2C, 3C, 25, 30, h = 12 A x² + y°, z? 1 1 1 1 1 1 A 1 1 -1 1 1 -1 R. E' -1 -1 (x, y) (x² – y², xy) A" A" 1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 E" -1 -2 1 (R,, R,) (zx, yz) 6 3 -2 2 -1 -2

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

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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Question

**Problem 1:**

**Objective:** Using the provided character table for the point group \(D_{3h}\), determine the irreducible representations contained in the reducible representation \(\Gamma\). Show your work.

---

**Character Table for \(D_{3h}\) (6m2):**

| Representation | \(E\) | \(2C_3\) | \(3C_2\) | \(\sigma_h\) | \(2S_3\) | \(3\sigma_v\) | Basis Functions |
|----------------|-------|---------|---------|--------------|---------|--------------|----------------|
| \(A_1'\) | 1 | 1 | 1 | 1 | 1 | 1 | \(x^2+y^2, z^2\) |
| \(A_2'\) | 1 | 1 | -1 | 1 | -1 | -1 | \(R_z\) |
| \(E'\) | 2 | -1 | 0 | 2 | -1 | 0 | \((x, y)\) |
| \(A_1''\) | 1 | 1 | 1 | -1 | -1 | -1 | |
| \(A_2''\) | 1 | 1 | -1 | -1 | 1 | 1 | \(z\) |
| \(E''\) | 2 | -1 | 0 | -2 | 1 | 0 | \((R_x, R_y)\), \((zx, yz)\) |

**Given:** \(h = 12\)

---

**Reducible Representation \(\Gamma\):**

\[
\Gamma = [6, 3, -2, 2, -1, -2]
\]

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