CH 612 Advanced Inorganic Chemistry Structure and Reactivity. Exam # 2 10/25/2010. Print name:

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1 CH 612 dvanced Inorganic Chemistry Structure and Reactivity Print name: 1. (25 points) a) Given the set of operations C 4, h determine the other operations that must be present to form a complete point group. If C 4 exists C 4 2 (= C 2 ), C 4 3 and C 4 4 (= E) also exist. C 4 h = S 4 C 2 h = S 2 = i C 4 3 h = S 4 3 b) Identify the point group for the complete set of operations. C 4h : E C 4 C 4 3 C 2 i S 4 S 4 3 h c) What is the order of the group? h = 8 d) Name at least three possible sub groups? Possible subgroups have g = 4,2,1. g = 4 C 4 {E, C 4, C 4 3, C 2 } S 4 {E, C 2, S 4, S 4 3 } C 2h {E, C 2, i, h } g = 2 C 2 {E, C 2 } C i {E, i } C s {E, h } g = 1 C 1 {E }

2 CH 612 dvanced Inorganic Chemistry Structure and Reactivity 2. (25 points) Complete the D 2h multiplication table below. Verify that any one of its reflection operations does not belong to the same class as any other reflection. D 2h E C 2(z) C 2(y) C 2(x) i (xy) (xz) (yz) E E C 2(z) C 2(y) C 2(x) i (xy) (xz) (yz) C 2(z) C 2(z) E C 2(x) C 2(y) (xy) i (yz) (xz) C 2(y) C 2(y) C 2(x) E C 2(z) (xz) (yz) i (xy) C 2(x) C 2(x) C 2(y) C 2(z) E (yz) (xz) (xy) i i i (xy) (xz) (yz) E C 2(z) C 2(y) C 2(x) (xy) (xy) i (yz) (xz) C 2(z) E C 2(x) C 2(y) (xz) (xz) (yz) i (xy) C 2(y) C 2(x) E C 2(z) (yz) (yz) (xz) (xy) i C 2(x) C 2(y) C 2(z) E The multiplication table is efficiently completed by considering the operator matrices for each operation and how they transform the x, y, z vectors : +z +y -x +x -y -z The operator matrices can be summarized as the reducible representation m D 2h E C 2(z) C 2(y) C 2(x) i (xy) (xz) (yz) m

3 CH 612 dvanced Inorganic Chemistry Structure and Reactivity Multiplication of each operation by the identity E results in the identical operation, e.g. [E][C 2(x) ] = = = [C 2(x) ] etc. Self binary combinations of each operation results in the identity operation E, e.g. [ (xy)][ (xy)] = = = [E] etc. The following binary combinations need then to be considered:

4 CH 612 dvanced Inorganic Chemistry Structure and Reactivity [C 2(z) ] [i] [C 2(y) ] [i] [C 2(x) ] [i] - - = = = - - = = = - - = = = [ (xy)] [ (xz)] [ (yz)] [ (yz)] [i] = = = [C 2(x) ] [ (yz)] [C 2(z) ] - - = = = [ (xz)] Likewise s the point group D 2h is an belian group all of the above combinations commute allowing us to easily fill the remaining of the multiplication table.

5 CH 612 dvanced Inorganic Chemistry Structure and Reactivity To verify that (xy) does not belong to the same class of (xz) or (yz) it similarity transforms must be derived using all of the symmetry operations of D 2h E (xy) E = E (xy) = (xy) C 2(z) (xy) C 2(z) = C 2(z) i = (xy) C 2(y) (xy) C 2(y) = C 2(y) (yz) = (xy) C 2(x) (xy) C 2(x) = C 2(x) (xz) = (xy) i (xy) i = i C 2(z) = (xy) (xy) (xy) (xy) = (xy) E = (xy) (xz) (xy) (xz) = (xz) C 2(x) = (xy) (yz) (xy) (yz) = (yz) C 2(y) = (xy) Thus, there are no other operations in the same class as (xy). 3. (20 points) Predict the representative character for the following combination of Mulliken symbols and symmetry class for the given point groups. point group Mulliken symbol class character (a) D 5d 2g i 1 (b) C 8 C 8 1 (c) D 2 2 C 2(x) 1 (d) C 6v 2 v 1 (e) D 3h 2 h 1 (f) C 6v E 2 C 3 1 (g) C 6v E 2 E 2 (h) C 5h h 1

6 CH 612 dvanced Inorganic Chemistry Structure and Reactivity 4. (20 points) Reduce the following representation into its component irreducible representations: Using the following formula we can carry out a systematic reduction to determine how many times (n) each irreducible representation contributes to the reducible representation. Thus, the reducible representation r is composed of the following irreducible components: r = E + 2 To check our answer the dimension of r must be calculated. i.e., d r = 5 which is the dimension of its character for the identity operation in r.

7 CH 612 dvanced Inorganic Chemistry Structure and Reactivity 5. Consider the following sequential structural changes (I II III). (30 points) M M M Indicate a) The point group of each structure. D 4h D 3h C 3v b) Whether structures I and II of each series bear any group subgroup relationship to each other. D 4h and D 3h bear no group subgroup relationship to each other. c) Whether structures II and III of each series bear any group subgroup relationship to each other. C 3v is a subgroup of D 3h. d) Whether each transition represents an ascent or descent in symmetry. descent D 4h D descent 3h C 3v ascent ascent (h = 16) (h = 12) (h =6) e) Specific symmetry elements lost or gained following the transition from II III. D 3h lost gained C 3v (h =12) (h =6) D 3h C 3v E, C 3, C 3 2, 3C 2, S 3, S 3 2, 3 v, h E, C 3, C 3 2, 3 v

8 CH 612 dvanced Inorganic Chemistry Structure and Reactivity f) Construct a correlation table(s) for any group subgroup pair present. D 3h C 3v x 2 + y 2, z z, x 2 + y 2, z 2 (x, y), (x 2 - y 2, xy) E 1 2 z 2 E (R x, R y ), (x, y), (xz, yz), (x 2 - y 2, xy) (R x, R y ), (xz, yz) E s the 1 representation in D 3h has no direct products associated with it, its correlating representation in C 3v must be identified using the characters of from each character table and their group subgroup relationship. D 3h E 2C 3 3C 2 h 2S 3 3 v C 3v x 2 + y 2, z 2 z, x 2 + y 2, z 2 (x, y), (x 2 - y 2, xy) z (R x, R y ), (xz, yz) E E E E (R x, R y ), (x, y), (xz, yz), (x 2 - y 2, xy) z, x 2 + y 2, z 2 (R x, R y ), (x, y), (xz, yz), (x 2 - y 2, xy)