9248 PM Tue Jul 6 "P G} 32% l a use i contemerowdmarkcom 4. The following questions refer to the coordinate system and schematic triangular gure (in blue) shown at right. The complete set of symmetry elements for this system is indicated in red; the av designations refer to mirror planes perpendicular to the xy plane and coplanar with the principal axis. Follow the labeling and rotational directions indicated in this gure. a. Group the symmetry operations into classes. b. Using the general Cartesian coordinates (am, y], 21) as the basis, write the 3x3 matrix representations for the following operations: E, C3, C32, (IV. C. The Cartesian matrix representations for the off-axis reections 0/ and a" can be derived by application of analytic geometry and/or vector algebra. Alternatively, these matrices can be obtained readily by visual inspection using the results of part (a) and the closure property of groups. Find these matrix representations using the latter approach. (Note: by convention, in a product sequence, matrix operators are applied from right to left to a column vector on the right.) (1. For the sequential operations Ov'\" Ov' C3, demonstrate the associative property of groups. You do not need to use explicit matrix representations; visual inspection and simplication of products is sufcient. Likewise, show that commutativity does not apply for the operations C3 and UV. 9:48 PM Tue Jul 6 32% usercontent.crowdmark.com operators are applied from right to left to a column vector on the right.) d. For the sequential operations o." .O. C3, demonstrate the associative property of groups. You do not need to use explicit matrix representations; visual inspection and simplification of products is sufficient. Likewise, show that commutativety does not apply for the operations C3 and ov. e . The matrix representations in parts (b) and (c) happen to be block diagonalized in irreducible form. What are the characters of their irreducible representations? f. Assign the conventional s, p, and d orbital basis functions to their respective irreducible representations. g. We can also choose the positions (A, B, C) as a basis set, as indicated in the figure above. Write matrix representations for all symmetry operations using this basis. What are the characters of this representation? Is this representation reducible or irreducible, and why? If the representation is reducible, find the component irreducible representations. PS # 5, p. 3 of 3 S.C.Lee 2021