A symmetry group of order h = 8 is known to have five (5) classes. Three of its irreducible representations have a dimensionality of 1. Based on this, what are the dimensions of its remaining irreducible representation(s)? 1, 1, 1 O 1,2 1, 1, 1, 1, 1 02, 2 1, 2, 2
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- Identify the symmetry elements present in the following objects. a The Eiffel Tower. You may have to look up a picture of it if you dont remember its shape b Any book ignore the printing. c An octagonal wood block. d A jack from the set of jacks pictured here: Note that some of the points end differently.Show that any two of the irreducible representations of the following point groups are orthogonal to each other. a C2 b C2v c D2h d Oh e TdIn your own words, explain why an object that has more symmetry elements is said to have higher symmetry than an object with fewer symmetry elements.
- Using the relationships imposed by the orthogonality theorem, predict the characters, XK(R), that have been omitted from this table, and match each with its correct value. All symmetry elements and irreducible representations (labelled with subscripts k = 1, 2, 3, 4, and 5) are included in this table. + X2(C3) ◆ X3(C₂) + X4(0d) X5(E) - ₁₁ [₂ [3 T4 [5 E 1 3 1 3 _?___ 6 S4 8 C3 1 1 1 -1 T -1 0 ? 1 0 -1 A. 1 B.-2 C. 2 D.-1 E. 3 F. 0 3 C₂ 1 -1 ? 어 -1 2 60d 1 -1 -1 ? OB) Identify all the elements of symmetry of IrCl(CO) [P(C6H5)3]2 and the point group assignment. PPH3 B CI- C=O Ph3PConsider the plane group p31m, explain: What is the detailed full meaning for p in this case? What is the detailed full meaning for 3 in this case? What is the detailed full meaning for 1 in this case? What is the detailed full meaning for m in this case? What symmetry elements are not explicitly listed in the plane group symbol but exist in the actual plane group unit cell? What is the orientation of each of all the symmetry elements identified in the previous question?
- Using its x, y, and z coordinates, construct the 15 x 15 matrix that describes the application of the S4 symmetry operation to square planar [PtCl 4]2- What is the character of this matrix?For the point group D3h, confirm that the irreducible representation E′ is orthogonal (in the sense defined by eqn 10B.7) to the irreducible representations A′1, A′2 , and E″.A scrambled character table for a polyhedral point group is shown here, though missing a couple of its characters. All symmetry elements and irreducible representations are included in this table. 6 S4 1 1 -1 -1 [₁ [₂ Гз T4 [5 What is the order of this point group? How many classes does this point group have? E 1 2 1 3 ? 8 C3 1 3 C₂ 1 -1 ? 1 0 −1 (report a positive integer) ? -1 2 6 Od 1 -1 -1 ? 0 (report a positive integer)
- Neopentane, C(CH3)4, has Tď symmetry, with character table provided below. Suppose the characters of a reducible representation of this point group are x(E) = 17, x(C3) = 2, x(C₂) = 5, x(S4) = -3, and x(od) = -5 (this is shown explicitly in the last row of the character table). (a) Use the T₁ irreducible representation to determine the order of the point group using the Little Orthogonality Theorem (Vallance Eq. 1.15.13, see Problem 3). (b) Determine how many times each irreducible representation in Ta is contained in this reducible representation. Td A₁ A₂ E Τ1 T₂ r E 1 1 2 3 3 17 8C3 1 1 -1 1 0 0 2 3C₂ 1 1 2 -1 6S4 1 -1 1 101 1.5 7 0 1 713 7. 1.5 -1 -1 60d -3 -1 -1 -5Å L. Y → X 5 3 Consider the trimethylenecyclopropane molecule (point group D3h). Each carbon atom has a pz orbital to contribute to the pi- MOs of the molecule. Take these 6 pz orbitais as a basis set and answer the following questions: A) How do they divide into subsets of symmetry - equivalent orbitals? B) What representations are spanned by each subset? C) What are the 6 normalizedThe molecule of NO2 belongs to the C, point group, with the C, axis bisecting the ONo angle. Taking as a basis the N(2s), N(2p) and O(2p) orbitals, identify the irreducible representations generated by these basis and construct the SALC's.