Chem 673, Problem Set 5 Due Thursday, November 29, 2007

Transcription

1 Chem 673, Problem Set 5 Due Thursday, November 29, 2007 (1) Trigonal prismatic coordination is fairly common in solid-state inorganic chemistry. In most cases the geometry of the trigonal prism is such that the ligand-ligand distances are nearly equal in the triangular and square faces of the prism. Just as you did in problem set #4 (problem 2) calculate the group overlaps of the appropriate SALCs for a trigonal prism (with equal edge lengths, a, as illustrated in the figure shown here) with the 5 d-orbitals. (Note the distance to the center of the triangular face is also supplied to help you with some trigonometry). Construct a d-orbital splitting diagram for a trigonal prismatic 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 MO 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 dl H Ld H LL E = E 15( )(0.15) 15( )(0.15) 2.5 E = E E = is the group overlap between the A 1 SALC (= 1 ( ) ) and the d z 2 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 lower energy solution 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 the problem. (2) Construct a d 2 state correlation diagram for a trigonal prismatic environment that is analogous to the diagrams given in Cotton in igures 9.3 and 9.4. Use your MO energy diagram from problem #1 to help get the strong-field limit side of your diagram. (3) Mn 3 is a high temperature vapor-phase species that could exist with either a D 3h, C 3v or C 2v geometry. The IR spectrum of Mn 3 matrix isolated in solid argon displays bands in the Mn- stretching region as follows 759 cm 1 (s); 712 cm 1 (m); and 644 cm 1 (vw). In the Raman spectrum only one band at 644 cm-1 could be observed above the noise level. (Note: although forbidden bands should not be observed, allowed bands are sometimes not observed.) (a) Determine the number of IR and Raman bands you would expect in the Mn- stretching region for each of these geometries. (b) Use this information to identify the molecular shape of Mn 3 and hence assign the observed vibrational modes to their symmetry species and sketch the form of the normal vibrations in each case.

2 (c) Propose an explanation for the observed geometry of Mn 3 (Hint: Consider the electronic structure of the alternative geometries; fluoride is a weak field ligand.) (4) (a) The infrared spectrum of matrix-isolated Cr 4 contains two bands, at 784 cm 1 and 303 cm 1. Consider four common geometries for the molecule: (i) tetrahedral, (ii) square planar, (iii) a partially flattened tetrahedron with θ > (see below) and (iv) a C 2v - symmetry, S 4 -like, structure. Decide which of these geometries consistent with the observed data. Cr (b) What would you expect to observe in the Raman spectrum of this molecule (assuming your answer to part (a) is correct). (c) What electronic states arise from the ground electronic configuration of Cr 4? (Make sure you have part (a) correct) (d) What electronic states arise from the electronic configuration of Cr 4 obtained by promoting one electron from the HOMO to the LUMO? (e) Referring to the states you found in parts (c) and (d), which HOMO LUMO transitions are dipole-allowed? (5) 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. (6) Slater determinants and Symmetries of States Hexanuclear rare-earth halide clusters are often open-shell molecules. My group has recently investigated the properties of [R 6 ZX 12 ] n L 6 clusters (R = Y, Gd, ; Z = Co, e, Mn; X = Cl, Br, I; L = neutral 2-electron σ-donor). These are octahedral clusters and it turns out that when central atom of the cluster is cobalt and the cluster charge is zero, the HOMO of the cluster has t 1u symmetry (the t 1u (z) orbital is shown in the illustration below; the t 1u (x) and t 1u (y) orbitals respectively exhibit an identical appearance if the molecule is viewed with the x or y axes are oriented vertically) and the three degenerate orbitals are occupied by 3 electrons; i.e., the O h symmetry cluster has a ground t 3 1u configuration. t 1u 3 t 1u (z) orbital

3 (a) ind all the states derived from the ground t 3 1u configuration of the cluster. This can t be easily done using character tables and methods given in Cotton s book, but methods discussed in class will work. (b) Only 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 O 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.] (c) How many Slater determinants can be constructed for which M S = 1/2? These determinants can be physically divided into two distinct types: (I) those which have all three electrons in different orbitals; (II) those in which two electrons are in the same orbital. Since there is no way for a symmetry operation to interconvert determinants of types I and II, it is useful to separate these two types. 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.] (d) or which irreducible representation(s) do determinants of type I form a basis? (e) or which irreducible representation(s) do determinants of type II form a basis? (f) In part (b), you dealt with the M S = 3/2 component of a quartet state, for which there are three other components with the same energy with M S = 1/2, 1/2, and 3/2. In either part (d) or part (e), you should have found which determinants are used to build the wavefunction for the M S = 1/2 component of this state. Use a projection operator for the quartet state symmetry to obtain a linear combination of determinants for the M S = 1/2 component of this state. (g) Now for some reasoning not based on symmetry. Shown at right is an approximate state (not orbital) diagram showing the relative energies of the states involved in this problem. The degeneracies of the states are not indicated. The two intermediate energies indicate distinct states that are accidentally degenerate (at the level of approximation used), and do not have 3K x,y the same energy by symmetry. K x,y is called an exchange integral, but physically arises from differences in the coulombic electron-electron repulsion in the states (its precise definition can be found in any good physical chemistry book). Using qualitative reasoning, explain which state is the lowest in energy and discuss which determinants should contribute most to the lowest energy states, the middle states, and the highest states? 2K x,y

4 Character formulas for all operations: (E) = 2J + 1 (C " ) = 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 wave function.

5 Some Useful Overlap Integrals Between Central-Atom s, p, and d Orbitals and Ligand and " Orbitals 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 ( 2 $ G 2 )S 2 S(xy, ) = 3GS S(xz, ) = 3HS 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 " = 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 or p z, d z 2, f xyz, etc. we use z, z 2, xyz, etc. c Ligand lies in the xz plane. or more general cases, a more complete table is needed. Reproduced from Burdett, Molecular Shapes, Table 1.1.