Chem 673, Problem Set 5 Due Thursday, December 1, 2005

Transcription

1 otton, Problem 9.3 (assume D 4h symmetry) Additional Problems: hem 673, Problem Set 5 Due Thursday, December 1, 2005 (1) Infrared and Raman spectra of Benzene (a) Determine the symmetries (irreducible representations) of the vibrational modes of benzene. (b) Which infrared fundamental bands are symmetry allowed? Raman? (c) Find the symmetries expected for the H and stretching regions of the spectrum. (d) Examine the H stretching region (~ 3100 cm 1 ) and offer an explanation for what is observed (is it expected?). (2) Some Seven-oordinate yanide complexes (a) Two important idealized geometries for y seven coordination are the monocapped x trigonal prism (a) and the pentagonal bipyramid (b). The seven-coordinate complex Mo() 4 Mo Mo 7 has been studied both as solid K 4 Mo() 7 2H 2 O and in aqueous solution. In the stretching region the (a) (b) IR spectrum has bands at 2119, 2115, 2090, 2080, 2074, and 2059 cm -1 for the solid and at 2080 and 2040 cm -1 for solutions. How many Raman and infrared bands would you expect for each geometry above and how many coincidences (bands present in both the IR and Raman spectra) should there be for each geometry? How do you interpret these data? (3) (a) Determine the number and activities of the carbonyl stretching modes of both the cis and trans isomers of L 2 M(O) 4. (b) Fe(O) 4 l 2 has IR bands at 2167, 2126, and 2082 cm -1 in Hl 3 solution. How would you interpret this spectrum? z

2 (c) Draw a picture of each carbonyl stretching symmetry coordinate of each isomer of L 2 M(O) 4. As an example of what we are looking for, the modes of mer-l 3 M(O) 3 are drawn above. The symbol stands for bond stretching and the symbol denotes bond compression. ote that the stretching of a unique chemical bond, as in the center diagram below, is always a legitimate basis for a symmetry coordinate. (4) The IR spectrum of Os(H 3 ) 4 ( 2 ) 2 2+ is shown in (a) below. Based on the appearance of the stretching region (~2000 cm -1 ), is this a cis or trans isomer? The IR spectra of I and II below are shown in (b). What is the formal oxidation state of Os in I and II? (Explain your answer!) an you interpret the spectrum of the stretching region in terms of local symmetry about the 2 groups? (Local symmetry means just considering nearest neighbors, i.e., X Y.)

3 (5) Slater determinants and Symmetries of States (a) Find all the states derived from the ground t 3 2g configuration of the r() 6 3- ion. This can t be easily done using character tables and methods given in otton s book, but methods discussed in class will work. [You can cheat and figure out the answer from inspection of a Tanabe-Sugano diagram. If you do this, please indicate that fact on your problem set and you must still go back and show how to do it without relying on the answer. In either case, explain how the answer can be obtained from a Tanabe-Sugano diagram.] (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 use of the O rotational symmetry group operations that your answer is correct. [Hint: In working this out, 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 different type, 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 y xz yz, where xy, xz, and yz refer to d orbitals, and the bar on top refers to a down-spin electron.] (d) For which irreducible representation(s) do determinants of type I form a basis? (e) For 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) ow 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 the same energy by symmetry. K xy,xz is called an exchange integral, but arises from a physical difference in the 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?

4 (6) Virtually all discussion of MR chemical shifts begins with Ramsey s expression, which breaks the chemical shift into two components, diamagnetic (σ d ) and paramagnetic (σ p ):! =! d +! p (1) L i"! " p (Z) = # µ 0 % L i" k k % 0 e2 i r % 3 i Z r k k 3 % L 0 i" i Z i 8$m % (2) 2 E k # E 0 k & 0 where Z is the nucleus under consideration, α runs over the artesian coordinates, k runs over all excited states, and i runs over all the electrons. The paramagnetic contribution to shielding, σ p, arises from the second-order mixing of paramagnetic excited states into the ground state by the applied magnetic field. The L iα operators are just the orbital angular momentum operators for the atom whose nucleus is under study (i.e., they are L x, L y, and L z for the α th atom). These operators transform in the same way as the rotations R x, R y, and R z. The denominators involve energy differences between the ground state (signified by " 0 " and " 0 " symbols in the numerator) and excited states, each labeled by the index k (signified by " k " and " k " symbols). Terms in the numerator involve integrals between ground and excited states (eg., k " L i! 0 ). i The presence of the paramagnetic term does not mean that the molecule being studied is paramagnetic (MR of paramagnetic molecules is often difficult to observe at all), but reflects the existence of low-lying paramagnetic excited states. σ p is by far the dominant contribution to the chemical shifts of most nuclei (but not those of hydrogen, 1 H, 2 H, 3 H) because chemical bonding for most atoms involves orbitals for which angular momentum is not zero (i.e., most atoms use p or d orbitals in bonding). [o(o 3 ) 3 ]3- [o(h 3 ) 6 ]3+ [o(acac) 3 ] [o(en) 3 ]3+ Mn(O) 5 o(o) 4 L i" [o() 6 ]3- [o(o) 4 ]- oh(pf 3 ) ppm [o(oh 2 ) 6 ]3+ op(o) 2 o 2 (O) 8 The 59 o nucleus has a very large MR chemical shift range; the figure above shows some representative complexes and their 59 o MR chemical shifts. All these complexes are low-spin and many are d 6 systems with approximately octahedral coordination environments.

5 Your problem: Write a clear explanation for the trend in the chemical shifts of the d 6 octahedral complexes in the figure. Once you have understood this problem, a clear explanation should be possible in one or two clear paragraphs. Your explanation should be as specific as possible and you should consider the following: (a) Ligand field strength (b) The restrictions that symmetry places on which integrals in the numerator of Ramsey s formula are nonzero.