Chem 673, Problem Set 5 Due Tuesday, December 2, 2008

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1 Chem 673, Problem Set 5 Due Tuesday, December 2, 2008 (1) (a) Trigonal bipyramidal (tbp) coordination is fairly common. Calculate the group overlaps of the appropriate SALCs for a tbp with the 5 d-orbitals assuming all the M-L distances are equal. Construct a d-orbital splitting diagram for a trigonal bipyramidal complex, assuming that the d- orbital energy, H dd, lies 2.5 ev above the ligand donor orbital energies, H LL. Perform a Hückel calculation on the complex, assuming that H dl for a single σ-donor (with overlap S σ ) is given by H dl = 15.0S σ ev and S σ = In other words, you will need to scale the interactions between the d- orbitals and the SALCs using the overlaps you calculate between the d orbitals and the relevant SALC and assume that S σ = [Note: recall that the d-orbital splitting diagram is just a part of the M diagram see the in-class discussion of octahedral coordination.] As an example of how to do this, the relevant secular determinant for the A 1 representation is as follows: H dd! E H dlax H dleq H dlax H Lax L ax! E H Lax L eq H dleq H Lax L eq H Leq L eq! E =!E!15S dlax!15s dleq!15s dlax!2.5! E 0!15S dleq 0!2.5! E = 0 where, for example, S dlax (= 2S σ ); 2 is the group angular overlap between the axial A 1 SALC (= 1 (! +! ) ) and the d z2 orbital. The zero of energy was chosen to be the d-orbital energy, and 2.5 is therefore the energy of the ligand based SALC. The two lower energy solutions of this will yield the primarily ligand-based bonding orbital; the higher energy solution will be the d z 2 orbital energy. You can solve this to get these energies and do the same for the other irreducible representations to finish this part. (b) Construct an orbital correlation diagram that connects the d orbital levels of an tbp ML 5 complex with those for a square pyramidal ML 5 complex (again assume the ligands, L, are σ-donors). The diagram depicted above describes the motion that interrelates these two geometries (known as a Berry pseudorotation after Prof. Steven Berry at the Univ. of Chicago). Note that the coordinate system used at the end is not the conventional coordinate system for the tbp, and though this does change the labeling of the Cartesian d orbitals for the

2 tbp, it doesn t change their symmetry, their form with respect to the ligands, or the d-orbital diagram. Assume that the motion is a pure bending motion of L 1, L 2, L 4, and L 5 ligands (all pivoting around the M-L 3 bond). Be sure to label the irreducible representations for the orbitals in the initial, final, and intermediate geometries for the pseudorotation shown. (Use the angular overlap table given to evaluate the d-orbital splittings.) (c) Assuming that only σ donor characteristics of the ligands are important, for what d-electron count(s) does the trigonal bipyramidal geometry seem most favorable? (2) Use group theory to determine (a) the allowed states for an (e g ) 2 configuration in D 3d. (b) the states into which the atomic 2 H state of d 3 configuration would split in D 3d. (3) Construct a d 2 state correlation diagram for a trigonal bipyramidal environment that is analogous to the diagrams given in Cotton in Figures 9.3 and 9.4. Use your M energy diagram from problem #1a to help get the strong-field limit side of your diagram. [In the problem that follows, some of your answers to later parts will depend of you getting earlier parts correct so be careful!] (4) The Fe(IV) ferrate ion, Fe 4 4, can be prepared by reaction of solid peroxides like Ba 2 with Fe 2 3. In impurity-free aqueous solutions, the Fe(VI) ferrate, Fe 4 2, is formed by disproportionation, 3Fe H 2! Fe Fe H x z Fe y The Fe 4 2 ion is tetrahedral and the Fe 4 4 ion has D 2d symmetry. (a) Give a brief explanation for the differing geometries of these two ions. A highly detailed answer is unnecessary. (b) Work out all the vibrational modes for the Fe 4 2 ion and indicate which of these modes are IR allowed and which are Raman allowed. (c) Work out the electronic ground state symmetry and all other states derived from the lowest energy configuration for the Fe 4 2 ion. Also, work out the states from excited configurations with the same spin multiplicity as the ground state. (d) Now consider the Fe 4 4 ion. If this ion were tetrahedral, work out the electronic ground state symmetry and all other states derived from the lowest energy configuration. Also, work out the states from excited configurations with the same spin multiplicity as the ground state. (e) Work out the symmetry of the vibrational mode (or modes) that describe the how the Fe 4 4 could distort to lower its energy (show how you arrived at this answer). (f) For the actual D 2d structure of the Fe 4 4 ion, all the Fe- bond distances are 1.81 Å, and the largest Fe angle is 126. Draw the structure of the Fe 4 4 ion; assume for the sake of simplicity that the distortion coordinate is a pure bending mode. Show that your

3 choice is consistent with your answer to part (e) by indicating which mode (or modes) was involved and draw the normal mode involved in the distortion; you can do this without any calculation if you use the symmetry characteristics of an orbital to help you see the mode by inspection. (g) Start with a d-orbital splitting diagram for a tetrahedral Fe 4 4 complex and construct an orbital correlation (Walsh) diagram in which the symmetry is lowered by moving along the distortion coordinate you drew in part (f). Using the appropriate subgroup, label the orbitals symmetries with the correct irreducible representations. (You must clearly label the coordinate system on the reduced symmetry molecule and make your irreducible representation labels consistent with your coordinate system and the character table below to receive credit!) (h) What is the electronic ground state symmetry of the distorted (D 2d ) Fe 4 4 ion? (i) Find all the IR and Raman active vibrational modes of the Fe 4 4 ion. (j) Find the ground state term ( 2S+1 L state) for a free Fe 4+ ion and all other states (if any) with the same spin multiplicity as the ground state. (k) Find the ground state term ( 2S+1 L state) for a free Fe 6+ ion and all other states (if any) with the same spin multiplicity as the ground state. (l) For just the state(s) with the same spin multiplicity as the ground state of the Fe 4 4 ion, construct a qualitative state correlation diagram, assuming this ion had a tetrahedral environment. (You can use results from parts d and j.) (m) Assuming a tetrahedral environment, work out the states for all the dipole and spin allowed d-d transitions for the Fe 4 4 ion that involve the orbital promotion of one electron. (n) For just the state(s) with the same spin multiplicity as the ground state of the Fe 4 2 ion, construct a qualitative state correlation diagram recall, this ion has a tetrahedral environment. (You can use results from parts d and k.) (o) Assuming a D 2d environment, for just the state(s) with the same spin multiplicity as the ground state of the Fe 4 4 ion, construct a qualitative state correlation diagram. (You can use results from parts h and k and some additional information you must derive.) (p) The structural information given in part (f) was derived from a solid-state x-ray structure determination it might be argued that the D 2d structure is the result of some asymmetric crystal packing forces. What observations would you expect to see in the electronic spectrum (UV-Visible-near IR) that would help establish that the D 2d structure persists in solution? (Explain.)

4 (5) Slater determinants and Symmetries of States (a) The graph below (right) has the same connectivity as the icosahedral-symmetry (I h ) molecule, C 60 (left). Draw such a graph (carefully) in the Hückel web calculator to get the π M diagram for C 60. Use the π bond orders of the two unique bonds and the Pauling bond order equation, d n = d log n (d 1 = single bond length = 1.54 Å for C C bond), to predict the bond lengths (Don t forget about the σ bonds!) (b) Construct the reducible representation for the pπ orbitals of C 60 and reduce it to work out the symmetries of all the pπ Ms. (c) Use the results from part (b), orbital pictures calculated in part (a), and/or reference to the literature to assign the irreducible representations to the orbitals. (d) The LUM of C 60 is important in its chemistry; the molecule is rather easily reduced and the ions C 60 n with n = 1, 2, 3, 4, 5 have all been prepared electrochemically in reversible reduction events. The C 60 ion is present in the superconducting salts A I3C 60 (A I = Na, K, Rb, Cs) which are of interest for their unusually high superconducting transition temperatures (up to 38 K). The remainder of this problem concerns the free C 60 ion. (e) Find all the states derived from the ground configuration of the C 60 ion. (If you ve done the previous parts of this problem correctly, you should be dealing with the t 3 1u configuration of the C 60 ion.) This can t be easily done using character tables and methods given in Cotton s book, but methods discussed in class will work. (f) nly one Slater determinant (label it as D 1 ) can be constructed for this configuration for which M S = 3/2. To what state does this determinant belong? Show by operating on D 1 with the I rotational symmetry group operations that your answer is correct. [Hint: Recall that the symmetry operators change only the spatial coordinates of the electrons and have no influence on the spins. Also remember that a determinant changes sign when any two columns are permuted.]

5 (g) How many Slater determinants can be constructed for which M S = 1/2? Write each of these determinants out (in compact form) and label them D 2, D 3, [Hint: To get you thinking on the right track, one the determinants of type I is x yz, where x, y, and z refer to t 1u orbitals that transform like x, y, and z, and the bar on top refers to a down-spin electron.] (h) For which irreducible representation(s) do the determinants listed in part g form a basis? Character formulas for a 2J +1-fold degenerate representation (for all operations):!(c " ) =!(E) = 2J + 1 sin[(j + 1 2)"] sin(" 2)!(i) = ±(2J + 1) sin[(j + 1 2)(" + # )]!(S " ) = ± sin[(" + # ) 2]!($ ) = ±sin[(j + 1 2)#] In these formulas E, C, S, σ, and i refer to the identity, proper rotation, improper rotation, reflection, and inversion operations, respectively. The angle of rotation is α. + signs apply for a gerade atomic state and signs apply for an ungerade atomic state, whether the point group under consideration has inversion symmetry or not. Examples: a 3 P state derived from either p 2 configuration (u u) or a d 2 configuration (g g) give g states. However, a 2 D state derived from a p 3 configuration (u u u) gives a u state, while a 2 D state derived from a d 3 configuration (g g g) gives a g state. The symbol J refers to the angular momentum quantum number of the state under consideration. In the Russell-Saunders scheme, J can be replaced by L when considering a spatial wavefunction.

6 Some Useful verlap Integrals Between Central-Atom s, p, and d rbitals and Ligand! and " rbitals a,b S(s,! ) = S! S(s," S(z,! ) = HS! S(z,"! ) = IS " S(z," # S(z 2,! ) = 1 (3H 2 $ 1)S 2! S(x 2 $ y 2,! ) = 3 (F 2 $ G 2 )S 2! S(xy,! ) = 3FGS! S(xz,! ) = 3FHS! S( yz,! ) = 3GHS! S(z 2,"! ) = 3HIS " S(z 2," # z! r L S(x 2 $ y 2,"! ) = $HIS " S(x 2 $ y 2," # S(xy,"! S(xy," # ) = IS " x " y S(xz,"! ) = (I 2 $ H 2 )S " S(xz," # S( yz,"! S( yz," # ) = HS " F = sin% cos& G = sin% sin& H = cos% I = sin% a "! is a ligand " orbital with an axis lying in a plane containing the z-axis and the ligand; " # is a ligand " orbital with an axis perpendicular to this plane. b For p z, d z 2, f xyz, etc. we use z, z 2, xyz, etc. c Ligand lies in the xz plane. For more general cases, a more complete table is needed. Reproduced from Burdett, Molecular Shapes, Table 1.1.