Assignment 2 Due Thursday, February 25, (1) The free-ion terms are split in an octahedral field in the manner given in this table.

Transcription

1 Assignment Due Thursday, February 5, 010 (1) The free-ion terms are split in an octahedral field in the manner given in this table. Splitting of the Free-Ion terms of d n configurations in an Octahedral Field Free-Ion Term Terms in Oh S P D F G H I A1g T1g Eg + Tg Ag + T1g + Tg A1g + Eg + T1g + Tg E1g + T1g + Tg A1g + Ag + Eg + T1g + Tg Formulas that allow one to derive these results are given below. χ(e) + 1 ; χ(i) = ±(J + 1) χ(c α ) = sin[(j + 1 )α] sin(α ) sin[(j + 1 )(α + π )] χ(s α ) = ± sin[(α + π ) ] χ(σ ) = ±sin[(j + 1 )π] 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. Eamples: a 3 P state derived from either p configuration (u u) or a d configuration (g g) give g states. However, a D state derived from a p 3 configuration (u u u) gives a u state, while a 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. (a) Verify the results given for the D and H states. (b) For all the states, add two more columns to the table for D4h and D3h point groups. (You can sometimes save work by choosing an appropriate subgroup.) () For each of the following pairs of molecules and ions, sketch a d-orbital splitting diagram for the parent species and the substituted derivative. Draw the diagrams for each pair side-by-side and show how the orbital energies are epected to change on going from parent to derivative. (Eplain your answers.)

2 (a) [PtCl4] trans-ptcl(nh3) (b) [RuCl6] 4 [Ru(CO)Cl5] 3 (c) [Co(NH3)6] 3+ trans-[co(nh3)4cl] + (3) UV-Visible spectra of two Ni(II) amide complees, [Ni(DMF)6] + and [Ni(DMA)6] +, are shown on the penultimate page. (DMF = N,N-dimethylformamide, DMA = N,Ndimethylacetamide) Using the attached Tanabe-Sugano diagram and the spectra, calculate o (= 10 Dq) and the Racah parameters, B, for each comple. Discuss the results in terms of the nephelauetic effect and the position of the ligands in the spectrochemical series. (Note that the energy scale is nonlinear; use the wavelength scale and convert to cm 1.) (4) Visible spectra of the OsX6 anions are shown on the net page. Corresponding regions of the spectra of the [Os(py)4X] complees are almost indistinguishable from one another. Eplain these observations. (5) This problem is intended to epose a little of the guts of ligand field theory. We will calculate the energies of the triplet states of an octahedral d 8 comple. Energy epressions for two of the triplet states, the ground 3 Ag(t 6ge g ) state and the ecited 3 Tg (t 5ge 3 g ) state, are easily obtained because they are the only triplet states with these symmetries. (Part a below.) However, the 3 T1g(t 5 ge g 3 ) and 3 T1g(t 4ge g 4 ) ecited states are of the same symmetry, and they mi to an etent that varies as a function of the ligand field strength. In the weak field limit, these two states descend from the free-ion 3 F and 3 P states and the energies are easily evaluated. However, the wavefunctions for the highfield 3 T1g states can be used as a basis for the zero-field free-ion 3 F( 3 T1g) and 3 P( 3 T1g) states. In doing this, we take advantage of the sum rule (see part d): the sum of the diagonal elements of the Hamiltonian (the trace) is equal to the sum of the energies, irrespective of the basis. This means that if we were to look at the high-field problem using the atomic 3 F( 3 T1g) and 3 P( 3 T1g) wavefunctions, the wavefunctions mi to yield the molecular wavefunctions, but the sum of their energies is unchanged (when the ligand field mies them, the upper state moves up eactly as much as the lower state moves down). Conversely, if we use the molecular 3 Tg(t 5 ge g 3 ) and 3 T1g(t 4ge g 4 ) wavefunctions to write the atomic Hamiltonian, the sum of diagonal Hamiltonian matri elements is equal to E( 3 F) + E( 3 P).

3 (a) As pointed out in the preceding (and as is evident in the d 8 Tanabe-Sugano diagram), the 3 Ag ground state and 3 Tg ecited state are the only triplet states of their respective symmetries. Consequently, you should be able to write a single-configuration wavefunction (for a MS = 1 spin component) for each of these states. (This is trivial for the ground state. For the ecited state, use the the descent in symmetry method discussed for Co(III) and Cr(III) complees in class.) By use of methods discussed in class, write down epressions for the energies using these wavefunctions and the formulas given at the tables attached to this homework set. Your answers should involve Racah parameter(s) and o (= 10 Dq). What is the energy difference between the two states? (b) Using the tables provided, find epressions for the energies of the free-ion 3 F and 3 P states in terms of Racah parameter(s). What is the energy difference between the two states? (c) Write a single-configuration wavefunction epression (for the MS = 3 / spin component) for each of the high-field 3 T1g states. (Just to save space, let s call these two states 3 A T 1 g.) These are obtained by assuming the only configuration that 5 contribute to these states are t ge 3 g (for 3 A 4 T 1 g) and t ge 4 g (for 3 B T 1 g). Once again, use the descent in symmetry method. Write down epressions for the energies using these wavefunctions in terms of Racah parameter(s) and o (= 10 Dq). What is the energy difference between the two states? (d) When two orthogonal basis states, Ψp and Ψq, interact with each other via a matri element Hpq, the appropriate secular determinant is H p E H pq H pq H q E = 0, which has solutions E = H + H p q ± ± H p H q + H pq. This has two limiting cases that are useful: (i) H p = H q, for which E ± = H p ± H pq, and (ii) H p H q H pq, for which E ± H p or H q Note that in every case the sum of the two energies is the same: E + + E = Hp + Hq. Let s use this information to work out the way that the two states, 3 A T 1 g, mi with each other over the entire range of ligand field strengths. We have already eamined the high-field limit in part (c); we have two 3 T1g states for which we know both the energies. Now, using the methods illustrated in class and the formula sheet attached on this problem set, find epressions for the free-ion 3 F and 3 P states. Starting with the energies of high-field 3 A T 1 g wavefunctions at zero field ( o = 0), deduce the off-diagonal matri element between these states that yields the correct energies of the atomic states, E( 3 F) and E( 3 P). (e) The off-diagonal matri element you found in part c applies at all ligand-field strengths, ( o 0). Therefore, you can write a general secular determinant in

4 the basis of the molecular 3 A 5 T 1 g(t ge 3 g ) and 3 B T 1 g(t 4ge 4 g ) wavefunctions that will give the energies of the 3 A T 1 g states for all ligand-field strengths. Solve the secular determinant to determine a general epression for the energies of the two states. (f) Using a spreadsheet program, plot the energies of the three ecited triplet states relative to the ground 3 Ag state on the horizontal ais, plot the energies as a function of o/b, and on the vertical ais, plot energies in units of B. In other words, plot three functions: E( 3 Tg) E( 3 Ag), E( 3 A T 1 g) E( 3 Ag), and E( 3 B T 1 g) E( 3 Ag). Do the results look familiar? (g) Etra Credit: The high-field wavefunctions, 3 A 5 T 1 g(t ge 3 g ) and 3 B T 1 g(t 4ge 4 g ), are a combination of the atomic wavefunctions, Ψ( 3 F) and Ψ( 3 P): Ψ( 3 T 1g A ) = c PA Ψ( 3 P) + c FA Ψ( 3 F) ; Ψ( 3 T 1g B ) = c FA Ψ( 3 P) c PA Ψ( 3 F) These are already constructed to be orthogonal and they must be normalized, c PA + cfa = 1. Find the relative 3 F and 3 P contributions to the high-field molecular states: cpa/cfa.

5 Relations Involving Coulomb and Echange Integrals p orbitals definitions J i, j = ϕ * i (1)ϕ * j ()( 1 r 1 )ϕ i (1)ϕ j ()dτ 1 dτ ; J 0,0 z,z, y, y J 1,1 1, 1 1,1 = ( i, j = ϕ * i (1)ϕ * j ()( 1 r 1 )ϕ i ()ϕ j (1)dτ 1 dτ 1 )(J, + J, y ) J 1,0 1,0, y,z y,z 1, 1 1,1, y, J, y 1,0 1,0, y,z y,z d orbitals J 0,0 z,z J,,, = ( 1 )(J y,y + J y ),y J,1, 1, 1,1 y,z J,0,0 y,z J 1,1 1, 1 1, 1 = ( 1 )(J z,z + J z, yz ) J 1,0 1,0 z,z 1, 1 z, yz z,z J z, yz, y, y y,y J y, y,1, 1 y,z ϕ z (1)ϕ y ()( 1 r 1 )ϕ yz (1)ϕ y ()dτ 1 dτ, 1,1 y,z + ϕ z (1)ϕ y ()( 1 r 1 )ϕ yz (1)ϕ y ()dτ 1 dτ,0,0 y,z 1,0 1,0 z,z Energies of real d orbital integrals in terms of Racah parameters J y,y z,z yz, yz z,z y, y A + 4B + 3C J z, yz y, yz y,z y, yz y,z A B + C J y,z y,z A 4B + C J yz,z z,z A + B + C J y,y A + 4B + C y, yz z, yz y,z y, yz y,z 3B + C y,z y,z 4B + C yz,z z,z B + C y,y ϕ z (1)ϕ y ()( 1 r 1 )ϕ yz (1)ϕ y ()dτ 1 dτ 3B For 1 st -row transition metals, Racah parameters B and C have typical ranges: B cm 1, C cm 1. (State energy differences don t involve A.) C

6 Energies of comple d orbital integrals in terms of Racah parameters J 0,0 J,,, J,1, 1, 1,1 J,0,0 J 1,1 1, 1 1, 1 J 1,0 1,0 1, 1,,1, 1, 1,1,0,0 1,0 1,0 A + 4B + 3C A + 4B + C A B + C A 4B + C A + B + C A + B + C 6B + C C 6B + C C 4B + C B + C The "Slater-Condon parameters" are defined by k r 1 F k e k r r 0 1 r < 0 k +1 r R (r ) R nl 1 nl (r ) k k +1 dr > dr ; r < 1 k +1 r = r if r > r 1 k > r k +1 r if r > r 1 1 and (in the d-shell): F 0 F 0, F = F 49, F = F The "Racah Parameters" are related to the Slater-Condon parameters by A = F 0 49F 4 B = F 5F 4 C = 35F 4

7