Chem 634, Assignment 2 Due Tuesday, February 27, 2018

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1 Chem 634, Assignment Due Tuesday, February 7, 018 The following two questions were given on the January 018 cumulative eam. They are not assigned here, but you should be able to fully answer them for study purposes: (A) The ligand field splitting parameters for octahedral complees are normally considerably larger than for tetrahedral complees, o > t. Nevertheless, the energy of the first d-d transition ( 6 A1 4 T) of Fe III O6 chromophores is around 11,000 cm 1, but is around,000 cm 1 for Fe III O4 chromophores. Give a concise but complete eplanation. (B) Give eplanations for the following: Assigned problems: a. [FeF6] 3 is colorless and [Fe(CN)6] 3 is red b. Sodium vanadate, Na3VO4, is a colorless solid and freshly prepared aqueous solutions of the VO4 3 ion are colorless. Potassium chromate, KCrO4, is an intensely colored yellow solid and yields deeply colored aqueous solutions (even when basic, where the CrO4 predominates over the CrO7 ion). Solid potassium permanganate is a dark purple solid and dissolves to form dark purple MnO4 solutions. The VO4 3, CrO4, and MnO4 ions all have d 0 configurations; what accounts for the color differences? (1) Electronic absorption spectroscopy can be useful for monitoring the course of reactions. In this problem you are asked to read through problem 5-59 in Harris and Bertolucci and to work out for all parts answers in detail. (Since short answers are given in the back of the tet, this is a gift! Because you will be responsible for knowing this material in the future, you are advised to work it through before looking at the answers.) You are epected to work independently on this problem. () Do problem 5-60 in Harris and Bertolucci and to work out the answer in detail the same comments apply! (3) 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. Use the spectrochemical series to guide you; recall that CO is a π acceptor and the oo ligand is a strong π donor. (Eplain your answers.) (a) [Co(NH3)6] 3+ trans-[co(nh3)4cl] + (b) [PtCl4] trans-ptcl(nh3) (c) Mo(CO)6 trans-[moo(co)4] (the d-count changes) (d) [Mo(CN)6] 3 cis-[moo(cn)4] (the d-count changes)

2 (4) Visible spectra of three Cr(III) complees are shown on the penultimate page. 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. (5) 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 O h S A 1g P T 1g D E g + T g F A g + T 1g + T g G A 1g + E g + T 1g + T g H E 1g + T 1g + T g I A 1g + A g + E g + T 1g + T g Formulas that allow one to derive these results are given at the top of the net page. (a) Verify the results given for the G and H states. (b) For all the states, add another column to the table for the D4h point group. (You can sometimes save work by choosing an appropriate subgroup.) The following formulas may be useful: χ(e) + 1 ; χ(c - ) = sin[(j + 1 )α] sin(α ) ; χ(i) = ±(J + 1) χ(s - ) = ± sin[(j + 1 )(α + π)] sin [(α + π) ] ; χ(σ) = ±sin[(j + 1 )π] In these formulae E, C, S, s, and i respectively refer to the identity, proper rotation, improper rotation, reflection, and inversion operations. The angle of rotation is a. + 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.

3 (6) The absorption spectra shown on the following page (and the notes) are for trans[crf(en)]clo4. Your task is to apply the same kind of the detailed treatment of the [transcocl(en)]+ ion given in class (including corrections!) and in Cotton s tet (Chemical Applications of Group Theory) to this case. Download and read the paper in which this data was reported (Inorg. Chem., 9, 188 (1970).) Green: 95 K; black: 4K. (a) What are the symmetries of the epected spin-allowed d-d transitions for this D4h ion? (Derive them all.) Show how these electronic transitions descend from transitions in an Oh ion. (b) What vibrational modes of the CrN4F core can possibly be involved in vibronic coupling? (c) Which transitions are vibronically allowed for z- or y-polarized light? (d) When this paper was reported, there had been no published crystal structure determination for trans[crf(en)]clo4 to allow these investigators to know, a priori, how the polarization of the incident radiation was related to the molecular orientation of the [CrF(en)]+ comple in the crystal. Nevertheless, the authors confidently suggest that the bands at 1.7 and 9.3 kk are due to the 4Bg 4B1g and 4Ag 4B1g transitions ( kk is an old notation, 1 kk = 1 kilokayser = 1000 cm 1). Fully eplain the basis of their confidence. (e) To the etent possible, assuming the authors confidence is justified, fully eplain their assignment for the four transitions at energies below 35,000 cm 1 in this figure. Be clear about how these transitions are related to analogous transitions for an octahedral comple (i.e., show how they descend from the transitions you can deduce from the Tanabe-Sugano diagram.) (f) For the ground state and each of the 4Ag and 4Bg ecited states, it is possible to write a single-configuration wavefunction epression (for the MS = 3/ spin component) which can be written in graphical form for each state with spin-up arrows in particular orbitals. Since the descent of each of these states can be traced back to Oh symmetry state(s), it is easy to get an epression for the energy difference between these states in an Oh ion. Using the methods illustrated in class and the formula sheet attached on this problem set, find epressions for the transition energies from the ground state to the first two spin-allowed ecited states for an Oh Cr(III) ion. Your answers should involve Racah parameter(s) and o (= 10 Dq). What is the energy difference between the two ecited states? (g) One of the transition energy epressions you found in part (f) is an overestimate which one is it? [Hint: Look at all the quartet states in the Tanabe-Sugano diagram.] (h) For the D4h comple, derive epressions for the transition energies from the ground state to the 4Ag and 4Bg ecited states. Your answers should involve Racah

4 parameter(s) and d-orbital energy differences (supply a molecular orbital diagram for the D4h comple with those energy differences clearly labeled). What is the energy difference between the two ecited states? (i) Hitchman is a coauthor of this paper, but in the more recent book Ligand Field Theory and its Applications, he gives rather different values for the Racah parameters B for [Cr(en)3] 3+ and [CrF6] 3 : they are respectively 639 cm 1 and 896 cm 1. These values suggest that the B value these authors used for trans-[crf(en)] +, 65 cm 1, may be far too small. If the weighted mean of the two prior values, 75 cm 1, had been used, how might the conclusions the authors reached concerning the relative s- and π-donor strengths of F and en be changed? (7) This problem is intended to give you a taste of the guts of ligand field theory. We will calculate the energies of the quartet states of an octahedral d 3 comple. (The results of problem 7 will be useful here.) Energy epressions for two of the quartet states, the ground 4 3 Ag(t g ) state and the ecited 4 Tg(t g e 1 g ) state, are easily obtained because they are the only quartet states with these symmetries (problem 7 and part a in this problem). However, the 4 T1g(t g e 1 g ) and 4 1 T1g(t g e g ) ecited states are of the same symmetry, and they will 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 4 F and 4 P states and the energies are easily evaluated. However, the wavefunctions for the high-field 4 A T g and 4 B T g states can be used as a basis for the zero-field free-ion 4 F( 4 T1g) and 4 P( 4 T1g) states. In doing this, we take advantage of the sum rule: 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 look at the highfield problem using the atomic 4 F( 4 T1g) and 4 P( 4 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 4 T1g(t g e 1 g ) and 4 1 T1g(t g e g ) wavefunctions to write the atomic Hamiltonian, the sum of diagonal Hamiltonian matri elements is equal to E( 4 F) + E( 4 P) (when the ligand field mies them, the upper state moves up eactly as much as the lower state moves down). (a) Proceeding just as you did in the last problem (parts f and g), write down epressions for the 4 3 Ag(t g ), 4 Tg(t g e 1 g ), 4 T1g(t g e 1 g ), and 4 1 T1g(t g e g ) states that appropriate for the strong ligand-field limit. Just to save space, let s call the last two states 4 A T 1 g and 4 B T 1 g. (b) When two orthogonal basis states, Yp and Yq, interact with each other via a matri element Hpq, the appropriate secular determinant is : H < E H <> H <> H < E : = 0, which has solutions E ± = H < + H > H < H > + H A <> This has two limiting cases that are useful: (i) H < = H >, for which E ± = H < ± H <>, and (ii) : H < H > : IH <> I, for which E ± H < or H >

5 Again, the sum of the two energies is the same in all cases: E + + E = Hp + Hq. Let s use this information to work out the way that the two states, 4 A T 1 g and 4 B 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 (a); we have two 4 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 4 F and 4 P states. (c) As indicated above, the wavefunctions for the high-field 4 A T g and 4 B T g states can be used as a basis for the zero-field free-ion 4 F( 4 T1g) and 4 P( 4 T1g) states. Starting with the energies of high-field 4 A T 1 g and 4 B T 1 g wavefunctions at zero field ( o = 0), deduce the offdiagonal matri element between these states that yields the correct energies of the atomic states, E( 4 F) and E( 4 P). (d) 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 the basis of the molecular 4 T1g(t g e 1 g ) and 4 1 T1g(t g e g ) wavefunctions that will give the energies of the 4 A T 1 g and 4 B 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. (e) Using a spreadsheet program, plot the energies of the three ecited quartet states relative to the ground 4 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( 4 Tg) E( 4 Ag), E( 4 A T 1 g ) E(4 Ag), and E( 4 B T 1 g ) E(4 Ag). Do the results look familiar? (f) Etra Credit: The high-field wavefunctions, 4 T1g(t g e 1 g ) and 4 1 T1g(t g e g ), are a combination of the atomic wavefunctions, Y( 4 F) and Y( 4 P): Ψ( L T NO P ) = c RP Ψ( L P) + c TP Ψ( L F) ; Ψ( L T NO V ) = c TP Ψ( L P) c RP Ψ( L F) These are already constructed to be orthogonal and they must be normalized, c RP A + c TP A = 1 Find the ratio of the 4 F and 4 P contributions to each of atomic states: c RP c TP A.

6 Relations Involving Coulomb and Echange Integrals p orbitals definitions J 0,0 z,z, y, y J i, j =!" * i (1)" * j ()( 1 r 1 )" i (1)" j ()d# 1 d# ; J 1,1!1,!1!1,1 = ( 1 )(J, + J, y ) K i, j =!" * i (1)" * j ()( 1 r 1 )" i ()" j (1)d# 1 d# J 1,0!1,0, y,z y,z K 1,!1!1,1, y,! J, y K 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 K 1,!1 z, yz z,z! J z, yz K,! y,! y y,y! J y,! y K,1!,!1 y,z! "# z (1)# y ()( 1 r 1 )# yz (1)#! y ()d$ 1 d$ K,!1!,1 y,z + "# z (1)# y ()( 1 r 1 )# yz (1)#! y ()d$ 1 d$ K,0!,0 y,z K 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 K y, yz z, yz y,z! y, yz! y,z 3B + C K y,z! y,z 4B + C K yz,z z,z B + C K! y,y "# z (1)# y ()( 1 r 1 )# yz (1)#! y ()d$ 1 d$!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

7 Energies of comple d orbital integrals in terms of Racah parameters J A + 4B + 3C 0,0 J,!,!,! A + 4B + C J,1!,!1,!1 A! B + C!,1 J,0!,0 A! 4B + C J 1,1!1,!1 1,!1 A + B + C J 1,0!1,0 A + B + C K 1,!1 6B + C C K,! K,1!,!1 K,!1!,1 K,0!,0 K 1,0!1,0 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

8 [Cr(NCS) 6 ] [Cr(urea) 6 ] [Cr(o) 3 ] λ (nm)

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