Molecular Spectra and Molecular Structure - 1987 PS3

Problem Set #3 1. Hund’s Coupling Cases. (a) Write the case (a) e and f –symmetry 3 × 3 effective Hamiltonian matrices for the 2 Π, 2 Σ+ problem we have considered in Lecture. Include only the zeroth and first order matrix elements of HROT and HSO . Show that the effective rotational constants for 2 Π3/2 and 2 Π1/2 are B ± B2 /A near the case (a) limit. (b) Consider the case (b) limit where A � BJ. Form the approximate case (b) eigenfunctions for 2 Π as �� � � �� ψ± = 2−1/2 ��2 Π1/2 ± ��2 Π3/2 and re-express the full 3 × 3 matrix in this basis. (i) You will find that both of the 2 Π eigenstates follow a BN(N + 1) rotational energy level expression. Which group of states (E+ , ψ+ ) or (E− , ψ− ) corresponds to N = J + 1/2 and which to N = J − 1/2? (ii) What is the ΔN selection rule for spin-orbit 2 Π ∼ 2 Σ+ perturbations? (iii) What is the ΔN selection rule for BJ·L 2 Π ∼ 2 Σ+ “L–uncoupling” perturbations? (c) Consider the case (c) limit for a “p–complex”. This means that the 2 Π and 2 Σ+ states correspond to � � the λ = 1 and λ = 0 projections of an isolated � = 1 atomic orbital. In this case 2 Π| BL+ |2 Σ+ = � � B[1 · 2 − 0 · 1]1/2 = 21/2 B. BΠ = BΣ = B, 2 Π|AL+ |2 Σ+ = 21/2 A, AΠ = A. Write the case (c) matrix and find the eigenvalues for EΠ = EΣ = E. What is the pattern forming rotational quantum number when A � BJ? For each J–value you should find two near degenerate pairs of e, f levels above one e, f pair. What is the splitting of these two groups of molecular levels? How does this compare to the level pattern (degeneracies and splitting) for a 2 P atomic state? (d) Consider the case (d) limit for a “p–complex”. Use the same definitions of EΠ , EΣ , BΠ , BΣ , AΠ , α, β as for case (c) but set A = 0. Your transformed case (b) matrix will be helpful here. Show that R is the pattern forming quantum number by finding the relation between R and J for each of the six same-J, e/ f eigenvalues. page 2 2. J. I. Steinfeld, p. 193, #2. The ground state and a low-lying excited electronic state of the BeO molecule have the following properties: Term symbol Electronic energy, T e /cm−1 ωe /cm−1 ωe xe /cm−1 re /10−8 cm 1 Σ+ 1Π 0. 1,487.3 11.8 1.33 9,405.6 1,144.2 8.4 1.46 Note that the electronic energy T e is the energy from the minimum of one curve to the minimum of the other; this is not equal to the vibrational origin of the 0 − 0 band. (a) Construct a Deslandres table of the vibrational band origins of the 1 Π − 1 Σ+ system, for v�� = 0 − 3 and v� = 0 − 5. Which of these vibrational bands would you expect to be the most intense when the system is observed in absorption? Comment on the relative intensities that you would expect for the other bands in your table. (b) In the rotational structure of the individual vibrational bands in this system, what branches would you expect to observe? In which branch would you expect to observe a band head? Identify the J � –J �� transition that will give rise to a line at the band head and the distance in cm−1 from the band head to the vibrational band origin. (c) What would you guess about the MO configurations corresponding to these two states? (Hint: Note that BeO is isoelectronic with C2 , so that the MOs may be expected to be somewhat similar, except that the g-u property will be lost, and the orbitals will be distorted toward the higher nuclear charge of the O atom.) Would you suspect the presence of any other excited electronic states below the 1 Π state? If so, what would its term symbol be? 3. J. I. Steinfeld, pp. 193-4, #3. The following bands are observed in the second positive system of nitrogen (units are reciprocal centimeters corrected to vacuum): 35,522 35,453 33,852 33,751 33,583 32,207 32,076 31,878 31,643 30,590 30,438 30,212 29,940 29,654 29,010 28,819 28,559 28,267 27,949 27,451 27,226 26,942 26,621 26,274 25,913 25,669 25,354 25,003 24,627 24,414 24,137 23,800 23,414 23,016 page 3 � �� � Arrange these in a Deslandres table, and find values for ω�� e , ωe xe , ωe , and ωe xe . (Important Suggestion: Look at the pattern of bands first, before doing anything else. Do any natural groupings seem to suggest themselves? It may help to draw a “stick spectrum” of the band origins, to scale, in order to pick out these patterns. Remember that bands having the same Δv fall along diagonals on the Deslandres table.) Is there any suggestion of a cubic term in (v + 1/2) in the vibrational energy level spacings of either electronic � �3 state? If so, derive an expression for the third difference, Δ3Gv+1/2 , including terms in ωe ye v + 12 in Gv+1/2 and estimate ωe ye . 4. J. I. Steinfeld, p. 195, #6. Draw two Morse curves defined by the following constants: T e�� = 0 cm−1 T e� = 10, 000 cm−1 −1 D�� e = 10, 000 cm D�e = 5, 000 cm−1 −8 R�� cm−1 e = 1.44 × 10 R�e = 1.54 × 10−8 cm−1 −1 ω�� e = 582 cm ω�e = 308 cm−1 and determine the two strongest transitions originating from v� = 19. Use a reduced mass of µ = 6.857 amu. Plot V � (R) − V �� (R) vs. R. Use this curve to determine: (a) The long and short wavelength limits of all bound-bound transitions in this system which possess signif­ icant intensity; (b) The long and short wavelength limits of strong bands from v� = 19; (c) Plot the R-value(s) sampled vs. transition energy for the progression of bands (v� = 19, v�� ) for v�� = 0 through the maximum v�� level that can be reached via non-negligible Franck-Condon factor from v� = 19. (d) The wavelength region(s) in which erratic intensity variations occur. Which v� levels participate in such transitions? Why? page 4 5. Effective Hamiltonian Matrices. (a) Set up the Hamiltonian, H = HROT + HSPIN−ORBIT for the 9 basis functions: 3Π 3 Σ+ Λ 1 −1 1 −1 1 −1 0 0 0 S 1 1 1 1 1 1 1 1 1 Σ 1 −1 0 0 −1 1 1 0 −1 Ω 2 −2 1 −1 +0 −0 1 0 −1 � �� �3 Π2 |vΠ � � �� �3 Π−2 |vΠ � � �� �3 Π1 |vΠ � � �� �3 Π−1 |vΠ � � �� �3 Π0 |vΠ � � �� �3 Π−0 |vΠ � � �� �3 Σ+1 |vΣ � � �� �3 Σ+0 |vΣ � � �� �3 Σ+−1 |vΣ � . Let α ≡ �Λ = 1�AL+ �Λ = 0� β ≡ �Λ = 1�L+ �Λ = 0� and use �1|H|2� = (−1)2J+S 1 +S 2 +σ1 +σ2 �−1|H| − 2� where σ = 1 for Σ− states and 0 for all other states, to ensure phase consistency for �Λ = 1|H|Λ = 0� and �Λ = −1|H|Λ = 0� matrix elements. (b) Construct the e/ f parity basis using the following phase definitions σv |v, nΛσ S Σ, ΩJM� = (−1) J−2Σ+S +σ |v, n − Λσ S − Σ, −ΩJM� e-levels σv ψ = +(−1) J ψ f-levels σv ψ = −(−1) J ψ. (c) Factor the 9 × 9 Hamiltonian into a 5 × 5 matrix and a 4 × 4 matrix using the e/ f basis functions. � � � � � � (d) Obtain the centrifugal distortion correction terms for the 3 Π0 |H|3 Π0 , 3 Π1 |H|3 Π1 , and 3 Π1 |H|3 Π0 e matrix elements. DΠ ≡ − � � � 2 � vΠ |B|vΠ vΠ� 0 − E0 EΠ Π� v v . e e page 5 (e) Obtain the correction terms for the effect of remote 3 Σ+ levels on 3 Π for the � � 3 Π |H|3 Π matrix elements. 1 1 � � 3 Π |H|3 Π 0 0 e and f and e and f � o≡ � v�Σ � �� 1 � 2 α v |v Π 2 Σ 0 0 EΠv − EΣv � � p≡4 � 1 2 αβ � �� �� vΠ |vΣ� vΠ |B|vΣ� 0 − E0 EΠ v Σv� � � ��2 � β vΠ |B|v�Σ v�Σ q≡2 v�Σ 0 − E0 EΠ v Σv� Express the diagonal contributions to the Λ–doubling of the Ω = 0 and 1 3 Π substates in terms of the o, p, and q–parameters.