1Chemistry 213a Advanced Ligand Field TheoryProblem SetsA. Weak Crystal FieldsB. Strong Crystal FieldsC. Intermediate Crystal Fields; Tanabe-Sugano DiagramsD. Introduction to Electronic SpectraTextsBallhausen, C.J. Introduction to Ligand Field Theory; McGraw-Hill, 1962.Figgis, B.N. Introduction to Ligand Fields; Wiley, 1966.Figgis, B.N. and Hitchman, M.A. Ligand Field Theory and Its Applications; Wiley, 2000.Griffith, J.S. The Theory of Transition Metal Ions; Cambridge University Press, 1961.Additional references have been placed on permanent reserve in the Millikan Library on theeighth floor.2IntroductionThe four accompanying problem sets will illustrate the development of crystal fieldtheory for the d3 electronic configuration in an octahedral ligand environment. In principle, oncethe calculations are understood for this case, they can be applied readily to other configurations.The theory of atomic spectroscopy provides the starting point for the crystal fieldformalism. For a many-electron ion or atom, the Hamiltonian has the form: ˆ H =h22m∇i2i∑−Ze2riji∑+e2riji>j∑+ξi(ri)lisii∑Here, the potential function takes into account interelectron repulsion and spin-orbit coupling inthe last two terms, respectively. For simplicity, the spin-orbit contribution will be ignoredinitially and considered later in the second half of this course (Ch 213b).The total energy of a dn term is given by the sum of its own electrostatic energy and itsenergy in the field of the core electrons. For our purposes, the core energy can be neglectedsince, to a first approximation, the core energy should be the same for all d3 configurations andwill vanish when energy differences between terms are considered.Upon complex formation, the Hamiltonian for the free ion will change to include theinteractions between the electrons and the ligand field, Vi, present in the complex. The strengthof the ligand field relative to other electronic forces present will vary from one complex toanother, allowing three principle cases to be identified:Vi<e2riji> j∑Vi=e2riji>j∑Vi>e2riji>j∑3These three cases correspond to weak, intermediate, and strong crystal fields, respectively. Theapproach for determining the nature of the electronic structure of a metal complex is first to findthe relative energies for the electronic terms of the system in each of these regimes and secondconstruct a correlation diagram that illustrates the change in relative term energies as the crystalfield varies from weak to strong (relative to the interelectron repulsion energy). The metalcomplex is then positioned on this diagram.The ligand field formalism follows a first-order perturbation treatment. In the weak-fieldcase, the crystal field is considered small compared to interelectron repulsions, while in thestrong-field case, the interelectron repulsions are considered small compared to the crystal field.For each case, wavefunctions will be constructed and diagonalized with respect to both energyoperators to yield energies for all the electronic terms in the complex. In addition to the ligandfield strength, the symmetry of the complex will affect the splitting of the terms. Therefore,group theory will play a critical role in simplifying calculations.Problem sets A and B will outline the procedure for determining energy expressions for ad3 ion in the limiting cases of weak and strong octahedral fields. In problem set C, the case ofintermediate ligand fields will be treated from both weak and strong-field perspectives. Thesetwo approaches will be shown to yield identical results and used to construct a simplifiedTanabe-Sugano diagram. Finally, problem set D will introduce some practical applications ofligand field theory to the electronic spectra of metal complexes.A partial list of references is included in this package. Several of the more useful textscan be found on the Ch 213 reserve shelf in the Millikan Library. You are encouraged to exploreany sources (including scientific journals) that may be helpful to you.4 A. Weak Crystal FieldsReading: Ballhausen (Intro. to LFT) Chapters 1-3, 4a-d, g; Gerloch and Slade (Ligand FieldParameters) Chapter 2; Weissbluth (Atoms and Molecules) Chapters 1 and 11; Figgis (Intro. toLF) Chapters 1-4.1. a.) Briefly explain the terms orbital, term, state, and level.b.) Using a microstate table, find all of the terms for the d3 configuration and determine theground state.c.) Using group theory, show how each term splits in the presence of a weak cubic field.2. Find the following wavefunctions by using the raising and lowering operators described inBallhausen:Ψ(3,2,32,12), Ψ(4,2,12,12), Ψ(5,2,12,12), Ψ(3, 2,12,12) 3. a.) Set up the equations to find the two wavefunctions for the state Ψ(2,2,12,12) . Do you seea problem? Do the two wavefunctions exist independently? Are they degenerate?Explain.b.) Detail a variational method for finding these wavefunctions.c.) Using a less rigorous approach, solve your equations in part a.) to find two wavefunctionsfor the 2D term. (Hint: arbitrarily set the coefficient of (2-,1+,-1+) equal to zero for onewavefunction.) How do these functions relate to your results from part b.)?4. Determine the energies of all the terms except 2D and 2P. For the quartet terms, use themethod described on page 23 of Ballhausen. Express your answers in terms of both Slater-Condon parameters and Racah parameters.55. a.) Derive an expression for the energies of the 2D terms in terms of two arbitrary,normalized, and orthogonal wavefunctions ψa and ψb where:)21,212, (2,b+)21,21(2,2,a)(baψψ=Ψ D2b.) Discuss this result in terms of the diagonal sum rule.c.) Find the mean energy of the terms. Part of this expression is needed in the next problem.Express your answer in terms of both Slater-Condon parameters and Racah parameters.6. Determine the energy of the 2P term.7. Determine the crystal field energies of all the quartet terms.8. a.) How would you go about finding the energies of the two 2D terms?b.) Find the energies of the wavefunctions that you derived in problem 3c. Solve for theenergies of the two 2D terms.9. Using the operator equivalent method, determine how the weak-field energies of the Oh statesarising from the 2F and 4F terms are related. Use this information to determine the ligandfield energies of the doublet term. (Refer to Griffith, Theory of Transition Metal Ions, page38 and Ballhausen, Intro. to LFT, page 84.)10. Like many