Chemistry 543--Final Exam--Keiderling May 5, pm SES

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1 Chemistry 543--Final Exam--Keiderling May 5, pm SES Please answer all questions in the answer book provided. Make sure your name is clearly indicated and that the answers are clearly numbered, indicating with which problems they are associated. Work must be shown for full credit. 'Short-cuts' can be used but should be noted for your own protection. Always remember that the set-up is worth the most in terms of partial credit. Calculators, rulers, pencils and molecular models permitted. If you need a specific fundamental constant, ask for it; but everything needed should be in the exam, unless I made an error! There is some possibly helpful information at the end of the exam. GOOD LUCK! I--Fundamental Quantum mechanics 1. To determine the observability of a spectral transition, we must first evaluate the matrix element (integral) of the operator, H', representing the perturbation due to the electromagnetic field that couples the ground or initial state, ψ g, to an excited state, ψ ex, i.e. <ψ ex H' ψ g >. Briefly discuss these questions: a. The source of the form of the perturbation we normally use is H'=(ieh/2π m)(a. ) where A=A 0 exp[-i(k. r-ωt)] is the vector potential. This is clearly a function of both space and time. The spectrum we measure does not change with time in general, eg. for absorbance of a stable molecular system. What happens to the time variable in our explanation of the interaction of radiation and matter? b. For atoms and molecules with centers of symmetry, the E1, electric dipole, selection rules are always different from the M1, magnetic dipole (and the E2, electric quadrupole) selection rules. Explain the source of this briefly. c. We have characterized our spectra (E1 selection rules) as having S=0 for allowed transitions. Explain how transitions that violate this rule can be observed. (Hint: There are two ways to change selection rules--change the operator or the state characteristics.) II. Atomic Spectra Consider the Ni +2 ion. (At # 28): a. What are the ground and first excited configurations for this ion. b. What state(s) arise from the ground configuration? What is the ground state? 1

2 (Expressions in term symbols, 2S+1 L, will suffice.) c. Are any transitions between these states from the ground state allowed by the E1 mechanism? Why? Is your answer different for the excited configuration? How? III. Diatomic spectra Consider HBr (At. Wt. H = 1.008, At Wt. Br = ): a. What rotational transitions can be detected in the microwave, the far-ir, and the Raman spectrum of HBr. (Selection rules specific to this molecule are needed.) b. Using the attached IR absorption spectrum of HBr, identify the quantum numbers of the ground and excited states for all degrees of freedom of interest (i.e. that change) in this transition for those labeled peaks: a, b, c, d. c. Determine the equilibrium bond length of HBr. If you wish to do a sample calculation, please be sure also to note the method of getting the "best possible" value for the ground state. d. Note why the transitions at higher energy are closer together than those at lower energies. e. What is the fundamental frequency of HBr? What is the force constant? IV. f. In the HCl spectra shown in class the peaks were all doubled due to the two isotopes of Cl, 35 and 37. Br has two isotopes, 79 and 81. Why aren't there doubled peaks? (extra credit!) Polyatomic spectra 2 Consider the molecule XeF 4 which is square planar in shape. This means that the Xe-F bonds are 90 apart, equal in length, and all lie in the same plane. Assume it has sufficient vapor pressure for all experiments noted below. (At. Wts.: Xe = 131.3, F =19.0, Cl =35.45) Rotational Spectroscopy:

3 a. What is the point group representing the shape of this molecule, XeF 4? that of XeF 3 Cl? b. Does this molecule have a permanent (equilibrium) dipole moment? Why? c. Classify this molecule as a " top" for purposes of rotational analysis. Be as specific as possible and defend your answer, i.e. make it clear that you know the definition of " ". Would your answer change for XeF 3 Cl? How? d. Identify the principal axes for XeF 4 and calculate the moments of inertia (I a, I b, I c ) about them. Assume that the Xe-F bond length is 2.3Å (0.23 nm). e. Evaluate the rotational constants: A, B, C. Draw an energy level diagram that includes the four lowest J values and any M J and K values corresponding to those J values. f. What are the electric-dipole-allowed microwave spectroscopic transitions for XeF 4 (in terms of J, M J, and K)? For XeF 3 Cl? Vibrational Spectroscopy: h. This molecule obviously also can vibrate. How many vibrational degrees of freedom do you expect? What are the symmetries of the normal modes corresponding to the vibrational degrees of freedom? Be sure that your answer is self-consistent. i. To which of these modes in part (h) are transitions allowed in the ir? to which in the Raman? How many fundamental vibrational transitions will be expected in each spectrum? Are there any overlaps? Any missing? Why? j. Which of these transitions are likely to be composed primarily of Xe _ F stretching motion? Be sure that your answer is complete and includes all the proper modes. 3

4 k. You can similarly determine the in-plane bending contributions from the F-Xe- F bends. What symmetries result? Do all of these correspond to real normal modes? Why? l. Are there enough internal coordinates to describe all the needed normal mode symmetries that you have identified? Explain any discrepancy, briefly. If some coordinate is missing from this approach, try to describe it by explaining what motion is involved. m. Assume Xe _ F stretches occur between cm -1 and XeF 2 bends at about 300 cm -1 (motions in sections j and k). Any modes from part l are probably between 200 and 300 cm -1. Sketch the expected IR and Raman spectra. Can you use a comparison of IR and Raman to assign symmetries to the states involved in these transitions? How? Do the polarization properties of either of the techniques discriminate between these vibrational transitions? How? Do the rotational selection rules help? How? n. In part (i and m) we addressed only fundamental transitions. What aspects of the Hamiltonian make possible the observation of overtones and combination bands? Consider just the XeF stretches from part j. Which first overtones will be allowed in the ir spectrum? Which combination bands? Where would these non-fundamental transitions be found in the spectrum? Would the Raman spectrum have different transitions? Why or why not? Electronic Spectroscopy: o. XeF 4 is a closed shell molecule with two lone-pairs. What is the multielectron ground electronic state symmetry? Explain briefly. Do not neglect spin! p. Working out the MO symmetries for XeF 4 would be a lot of work even in a minimal basis set. Let's just look at the "minimal valence set" composed of 8 orbitals total: the 5s and three 5p orbitals on Xe and either the 2p x or 2p y on each of the four F's which are chosen to point at the Xe. [Note these are the 4 orbitals that would form the valence σ bonds and are symmetry 4

5 interconverted by the C 4 operation.] Determine the representations of linear combinations of these eight orbitals and the shape of the secular determinant. Postulate an energy ordered MO diagram, labeling the orbitals by symmetry and according to bonding, non-bonding or anti-bonding character. Be sure to make your ordering consistent with a closed shell ground state by reordering the highest energy orbitals as necessary. Show interconnectivities to the AO's used as your basis. q. Part p was created to be a 12 e - problem (Xe orbitals filled, but F orbitals half filled); determine the lowest energy configuration based on your diagram. The state symmetry it generates should match part (o). Note that the answer is not unique but depends on your assumptions in part (p). Be sure to account for degeneracy. r. What are the HOMO (highest occupied MO) and LUMO (lowest unoccupied MO)? Do they fit your expectations from a simple valence bond picture of bonding electron pairs and lone pairs? Why or why not? s. What are the first and second excited configurations using your scheme? What are the state symmetries that these configurations develop? Be careful about spin with degenerate orbitals! t. Using the results of parts (q) and (s) determine what are the electric dipole allowed transitions (E1) and the magnetic dipole allowed transitions (M1). Make an energy level diagram of the states to indicate which transitions are allowed and by which mechanism and which are forbidden. u. Sketch what you believe will be the shape of the absorption spectrum for the lowest energy, E1 allowed transition in your diagram. Indicate the origin and magnitude of any vibrational spacing seen. There will be more than one possible answer. Be sure to give your reasoning. v. The E1 forbidden transitions are often vibronically allowed. Which forbidden transitions" will be vibronically allowed? Which of the vibrations can generate false origins" from which progressions will commence? Which vibrations will most likely create progressions? What is the expected spacing of the vibronic transitions? [For simplicity, just use the Xe-F stretches in part (j).] Describe any differences you would expect between the low temperature (0 o K) and the high temperature (~1000 o K) spectrum of this transition. 5

6 V. Group theory a. Determine the point groups of: allene, C 3 H 4 BrF5 Cu(NO 2 ) 6 4- Nd(C 5 H 5 NO) 8 4+ b. Use the projection operators to develop a symmetry adapted linear combination of 2p z orbitals on the C's, which can describe the π-orbitals formed by their overlap, for cyclobutadiene, C 4 H 4, which is assumed here to be a planar anti-aromatic molecule. Hint: First determine the point group, then the irreducible representations of the 4 2p z orbitals, then use the projection operators. Since cyclobutadiene is assumed to be anti aromatic, the double bonds will be shorter than the single bonds. Let z be perpendicular to the plane of the molecule. 2nd Hint: the character form, P Γ, is sufficient for solving this problem if you make a table of the consequences of each symmetry operation on the 2p z orbital, i.e. R. (2p z ) values for all R operations in the group. Use of P Γ will generate only one of the functions transforming as a degenerate representation. 3rd hint: you do not need all the operations in the group if a subgroup does the mixing of the functions, but the arithmetic is easier if you avoid cyclic groups. Hint 3: You can save work by determining the representations of the p z orbitals first. The calculation is quick if you see the patterns. 6