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Lecture 9 Problems - Winter 2006

Electronic Spectroscopy - Lectures 9a, 9b & 9c

Diatomic MoleculesExercise 9.1 (a) Which of the following transitions are electric dipole allowed: (i) 2Π-2Π, (ii) 1Σ-1Σ, (iii) Σ-∆,(iv) Σ+-Σ-, (v) Σ+-Σ+, (vi) 1Σg

+-1Σu+, (vii) 3Σg

--3Σu+? (Use the selection rules given in Lecture 9).

(b) Deduce all of the allowed electronic transitions allowed for (i) H2 and (ii) I2 for which thelower state is 3∆u.

Exercise 9.2 Construct the full Hamiltonian for a system of two nuclei (denoted A and B) and two electrons(denoted 1 and 2) in a diatomic molecule. Which terms are eliminated by the Born-Oppenheimer approximation? What about the SCF approximation?

Exercise 9.3(a) Write the ground configuration for each of the diatomic molecules below, sketch the MOdiagram, and fill in the following table:Molecule force constant (N m-1) bond order

NH 3133

NF 1115

N2 2331

O2 1555(b) For each set of molecules, which has the highest stretching frequency? Explain.(i) NO, NO+, NO- (ii) CP, CP+, CP- (iii) SiS, SiS-, SiS2-

Exercise 9.4 (a) Using the wavelength scale of the spectrum on the following page, measure, as accurately asyou can, the wavelengths of the R(5), P(5) and P(7) lines of the spectrum of RhN (at best, thesewavelengths would be accurate to ±0.001 nm - you can use a photocopier enlargement toimprove accuracy if you wish). Convert these values to wavenumbers, and use the appropriateequations to obtain B0N and B0O.(b) For the ground electronic state of 19F2, B0O = 0.88326 cm-1 and B1O = 0.86941 cm-1. Calculatethe value of αO and reO, given that m(19F) = 18.998 u.

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Exercise 9.5Sketch potential energy curves for the following states of CdH, Br2 and CH, given theirinternuclear distances, re, and suggest qualitative intensity distributions in the vO = 0progressions for transitions between the states observed in absorption.Molecule State re / ÅCdH X2Σ+ 1.781

A2Π 1.669Br2 X1Σg

+ 2.281

2.678

CH X2Π 1.120C2Σ+ 1.114

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Exercise 9.6Give the name (or symbol) for each of the 10 features below labelled with a question mark.

Exercise 9.7 Consider the A3Π - X1Σ+ electronic spectrum of 127I79Br. The leading vibrational constants for theexcited state are ωeN = 133.45 cm-1 and ωexeN = 0.609 cm-1, and a rotational constant BvN =0.04243 - 0.00064 (vN + ½). The leading vibrational constants for the ground state are ωeO =266.63 cm-1 and ωexeO = 0.817 cm-1, and a rotational constant BvN = 0.0568 - 0.00019 (vO + ½). For this electronic transition, ∆Te = TeN - TeO = 12,370.01 cm-1 (∆Te and E00 are both symbolsdescribing the energy difference between the ground vibrational states in the excited andelectronic states).(a) What is the band origin for the (vN,vO) = (2,4) band in this spectrum? The band origin iscalculated with υ0(vN,vO) = ∆Te + GN(v) - GO(v).(b) Sketch a hypothetical potential energy diagram assuming that reN is a bit larger than reO. Show the (vN,vO) = (2,4) and (vN,vO) = (0,0) transitions (the latter is the E00 transition).(c) What are the energies of the P(20) and R(10) lines in this band (remember that BvN and BvOare different in this case - see equation sheet on combination differences).

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Polyatomic MoleculesExercise 9.8 Show that for the carbonyl group in formaldehyde the π* 7 π transition is allowed, with thetransition moment along the bond.

Exercise 9.9 The fundamental vibrational frequencies of formaldehyde in the electronic ground state are asfollows: v1 = 2766.4; v2 = 1746.1; v3 = 1500.6; v4 = 1167.3; v5 = 2843.4; v6 = 1251.2 cm-1.(a) What is the total energy of the molecule in the ground electronic state at 0 K? (Answer incm-1 and call the minimum of the ground state potential well 0 cm-1). Suppose the energy of the0-0 electronic transition to the 1A2 (π-n) excited state is designated as E00. The vibrationfrequencies in the excited state are: v1 = 2847; v2 = 1173; v3 = 887; v4 = 124.6; v5 = 2968; v6 =904 cm-1.(b) What is the energy of the transition from the ground state (lowest vibrational state of thelowest electronic state) to the excited electronic state, 1A2, in which v3N is also excited by onequantum (i.e., the transition)?(c) Draw a potential energy diagram for the ground and first excited states, and show thetransition from the first excited vibrational state v4 in the electronic ground state to the groundvibrational state of the 1A2 electronic excited state. The transition is designated . What is theenergy of this transition?

Exercise 9.10 Show that the hot band transition 1Ag(v8O) 6 1B2u(v7N) of p-fluorobenzene shown below isvibronically allowed and z-polarized. The notation means that the molecule is excited from thev8 (b2g) vibrationally excited electronic ground state to an electronic excited state in which v7N(b1g) is vibrationally excited. The notation for the transition is .

Use the attached direct product table for D2h to save time in calculating direct products.

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Exercise 9.11 Show that the transition 1A2 7 1A1 is electric dipole forbidden in H2O, but may become allowed asa vibronic transition involving one of the vibrational modes of the molecule. Note that theground vibrational state always has symmetry of the totally symmetric representation. Use thedirect product table for C2v to rapidly evaluate direct products. (Lectures 15, 16, 21)

Exercise 9.12 (a) In a diamagnetic octahedral complex of Co3+, two transitions can be assigned to 1T1g - 1A1gand 1T2g - 1A1g. Are these transitions forbidden? If they are forbidden, what symmetries ofvibrations would provide intensity? Again, use the direct product tables.(b) Write down the crystal field configurations of the following t.m. complexes: (i) [Cu(H2O)6]2+,(ii) [V(H2O)6]3+, (iii) [Co(NH3)6]3+ (low spin) and (iv) [Mn(H2O)6]2+ (high spin). Label theground state for each case. (c) Explain why [Mn(H2O)6]2+ is colourless.

Exercise 9.13 Discuss, briefly, the valence MOs of AlH2 and the shape of the molecule in the ground and firstexcited singlet states. Sketch an MO diagram to accompany your discussion. Is an electronictransition allowed between these two states? Why or why not?

Exercise 9.14 Using Hückel theory, determine the energies of the π MOs of a) cyclobutadiene, b) butadieneand c) the cyclopentadiene radical (5 e- in pz AOs). Sketch the energy levels and rough diagramsof the MOs, labelling the ground state as singlet, doublet or triplet. Use Maple or an appropriateprogram or calculator if you wish to quickly evaluate the determinants and solve for x.

Exercise 9.15 In the first excited singlet state, S1, ethyne has a trans bent configuration with pHCC = 120o

compared to 180o in the linear ground state, S0. In addition, the C/C bond length increases to1.34 Å in S1 from 1.21 Å in S0. What kind of vibrational coarse structure do you expect for theS1-S0 absorption spectrum?

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Challenge Problems:Exercise 9.16 Electronic levels for O2 are shown below.

(a) What is the electron configuration of the X, a and b levels? What are the associated termsymbols? (see table at end of assignment)(b) What is the electron configuration and term symbols associated with the A and B states? Note: The B-X transition is the most intense (see below also).(c) The spin-forbidden transitions a-X and b-X for O2 occur in the atmosphere. In what part ofthe EM spectrum are these transitions observed?(d) Absorption and fluorescence are indicated by the arrows shown in the figure above. Whatare the approximate wavelengths of the photons involved?(e) A study of the Schumann-Runge bands for the B-X transition in O2 showed that the limitoccurs at 1759 Å. Given that the energy difference between O(3P) + O(1D) and 2O(3P) is15867.862 cm-1, determine D0 for the ground level of O2.

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Exercise 9.17Part of the MO scheme of the pyrazine molecule is shown below.

(a) Draw pictures of the n+ and n- lone pair MOs, based upon their symmetry species.(b) Write the ground state MO electron configuration of pyrazine.(c) Write the first two excited configurations arising from promotion of a single electron from theoccupied n+ and n- MOs to the π4* MO. What states arise from these new configurations? (Direct producttables and character tables are available in the supplementary information).(d) Evaluate the integrands of the transition moment integrals for the 0-0 bands (i.e., no associatedvibrational transitions) of the two lowest energy “spin-allowed” transitions (i.e., singlet-singlet). Thesetwo transitions are from the totally symmetric ground state to excited electronic states with the symmetryspecies determined in part (c). Is either transition orbitally (i.e., electronically) allowed? Identify thedirection of the transition moment for these transitions.(e) For only the orbitally-allowed transition(s) from part (d), what are the symmetries of vibrations thatcan serve as vibronic origins for transitions? Hint: evaluate the integrand in Rev where the groundelectronic and vibrational states are totally symmetric.(f) The vapour phase spectrum of pyrazine is shown on the next page. Two bands are labelled A and B. Assign A and B to one or other of the two transitions above, and justify your reasoning.(g) The peaks labelled with X, Y and Z are progressions. Specifically, the peaks labelled with Y1, Y2 andY3 are from a progression associated with peak B: the vibronic transitions . If υ5 is atotally symmetric stretch with a fundamental frequency of 516 cm-1, roughly sketch a energy leveldiagram showing the origins of this progression. Label the transitions B, Y1, Y2 and Y3, as well as E00.(h) Based on the intensity of the 0-0 transition and associated vibronic transitions, what do you expect tobe the relationship between reN and reO (i.e., equilibrium bond lengths in the excited and ground states). Explain using the Franck-Condon principle and some rough potential energy surface diagrams.

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Other Information:If the rotational constants are different in the ground state and first excited state:

Electronic configurations and term symbols in molecules: