Molecular Spectra and Molecular Structure - Lec 13S

Lecture #13 Supplement: Dreckium Monoxide Dreckium is a superheavy element which belongs in Group IIA of the periodic table. It has atomic number 120 and there are two isotopic species 324 Dk and 326 Dk with atomic weights 324.00 and 326.00 and relative abundance 95% and 5%. Recently, spectra of DkO have been reported. A. Microwave Spectrum (Recorded at T = 1500K) Microwave spectra have been obtained in the Ku (12-18 GHz), Ka (26-36 GHz), and Q (33-50 GHz) band regions. The following transitions and their relative intensities are observed. Intensity ν (MHz) . 16560.08 16555.30 16524.42 16519.66 16488.76 16484.02 33120.04 33110.48 33048.72 33039.20 32977.40 32967.92 49679.76 49665.42 49572.78 49558.50 49465.80 49451.58 100 5 50 3 30 1 100 5 50 3 30 1 100 5 50 3 30 1 It is not meaningful to compare relative intensities of widely separated lines. An extensive search of the regions near 8250 MHz and 24700 MHz yielded no additional lines. B. Infrared Spectrum Several bands were observed and the following vibrational constants for the ground state were obtained: 1 324 ωe ωe xe Dk16 O 326 Dk16 O 710.17 cm−1 710.07 1.02 1.02 2 C. Electronic Spectrum One band in the electronic absorption spectrum was recorded in which a head formed near 4387 Å. It was believed that lines of comparable intensity separated by 0.03 cm−1 could be resolved. The band was not rotationally ana­ lyzed because, although it appeared to be part of the A1 Π−X1 Σ+ band system, the computer automated assignment procedure was rendered ineffective by per­ turbations. The following lines and relative intensities were observed: 22794.48cm−1 94.40 94.16 93.75 93.62 93.29 93.20 92.81 92.49 92.16 91.63 91.34 90.61 90.37 89.48 89.42 89.23 88.23 88.08 87.93 86.89 86.57 86.46 85.50 84.91 84.84 84.58 2 4 4 4 2 4 4 5 7 7 8 8 8 9 5 4 10 5 4 10 4 5 11 3 5 11 1 84.13 83.63 83.10 83.05 82.75 82.46 81.40 81.12 81.02 80.02 79.30 79.00 78.58 77.32 76.78 76.72 75.09 74.43 74.28 72.66 71.98 71.68 70.78 70.04 69.48 68.92 68.72 3 1 5 11 2 2 1 16 3 1 3 16 0 3 5 10 3 5 9 4 4 9 1 3 3 8 1 67.24 67.01 66.44 65.99 64.53 64.25 63.90 62.90 62.07 61.08 59.66 59.59 57.99 57.78 57.05 56.24 54.66 54.29 52.66 51.13 48.93 47.40 45.03 43.49 40.97 39.41 35.15 30.73 3 2 2 8 1 3 2 7 1 5 6 1 3 2 0 5 3 2 5 3 4 2 4 2 3 2 2 1 In an effort to understand the perturbed band, a low resolution absorption spectrum of Dk18 O was recorded. Four red degraded bands were observed in the isotopic spectrum near 4387 Å: 22675.51 cm−1 22706.55 23110.51 23201.77 A1 Π − X1 Σ+ B1 Σ+ − X1 Σ+ A−X B−X 3 � (vΠ , v �� = 0) � (vΣ , v �� = 0) � (vΠ + 1, v �� = 0) � (vΣ + 1, v �� = 0) All four of these bands appear to be free of perturbations and the two B−X bands are much weaker than the A−X bands. 1. A. (10 points) Assign the microwave spectrum. B. (10 points) Determine Be , αe , De , and re for 324 Dk16 O and 326 Dk16 O. Be realistic about uncertainties of the derived constants. C. (5 points) �Use the Kratzer relation to estimate ωe for 324 Dk16 O: De = 4Be3 ωe2 . Now use the Pekeris relation to estimate ωe xe for � � 324 Dk16 O: αe = ω6e (ωe xe Be3 )1/2 − Be2 . 2. A. (30 points) Rotationally analyze the 4387 ˚ A band. [HINT: 82 lines are too many to attempt to assign without a specific plan of attack. I suggest the following: a. Prepare a long (∼ 1�� per cm−1 ) stick spectrum. Represent the relative intensities as stick heights. Be sure to leave about 1�� of space above and below your stick spectrum. You will use this space to organize the lines into branches using a “railroad tie” diagram. b. Prepare another long strip of graph paper using exactly the same length per cm−1 scale. Plot ground state combination differ­ ences, Δ2 F (J �� ) = B �� (4J �� + 2) − D�� (8J 3 + 12J 2 + 12J + 4), de­ termined from the microwave constants. Before making separate strips for several v �� levels, do some spot checking to determine the v �� assignment for the band. You are going to use this strip as a ruler to find P (J + 1) and R(J − 1) lines connected to a common J � level. The zero of the strip is placed on top of the P (J + 1) line and the J’th combination difference mark will then fall on top of the R(J − 1) line. P (J + 1) is always to the red of R(J − 1). Lines in a branch vary smoothly in intensity (unless two lines accidentally fall on top of each other) and the separation of consecutive members of a branch varies smoothly (except at some perturbations): c. Prepare the “railroad tie” diagram. As you recognize lines connected with a common upper level with known J � , mark these lines as follows: 25 R Q P 28 27 27 27 26 28 29 This should permit you to identify every line in the above list. Note that, at perturbations, there may be more than one upper level with a given J � value. End of HINT.] 4 B. (10 points) Obtain band origins and B values for the A1 Π and perturbing B1 Σ+ state. The easiest way to do this is by some sort of graphical method. Recall that in a 1 Π ∼1 Σ+ perturbation, the 1 Π f–levels are not affected by the perturbation. The perturbation matrix element, �1 � Πe |H|1 Σ+ = β �vΠ |B|vΣ � [2J(J + 1)]1/2 , is J–dependent. It will be necessary to convert observed transition frequencies into term energies (energies of the J� level above v �� = 0, J�� = 0) by adding to the transition frequency a suitable ground state rotational energy. You will need to prepare a table of such ground state rotational energies (perhaps at the same time that you are preparing your table of Δ2 F(J�� ) values). C (5 points) Describe how you know whether the perturbation in­ volves the upper or lower electronic state. What is the magnitude of the rotationless part of the perturbation matrix element, β �vΠ |B |vΣ �? What determines the relative intensities of main and extra lines? D II A. (10 points) Use the isotope shift information to obtain vibrational assignments of the A1 Π and perturbing level. You do not have sufficient information to determine both ωe and ωe xe ; you will have to guess a reasonable value for ωe xe . Effective Hamiltonian Matrices (15 points) Set up the Hamiltonian H = HROT + HSPIN−ORBIT for the 9 basis functions: 3 Π 3 Σ+ Λ 1 -1 1 -1 1 -1 S 1 1 1 1 1 1 Σ 1 -1 0 0 -1 1 Ω 2 -2 1 -1 +0 -0 |3 Π2 �|vΠ � |3 Π−2 �|vΠ � |3 Π1 �|vΠ � |3 Π−1 �|vΠ � |3 Π0 �|vΠ � |3 Π−0 �|vΠ � 0 0 0 1 1 1 1 0 -1 1 0 -1 |3 Σ+ 1 �|vΣ � 3 + | Σ0 �|vΣ � |3 Σ+ −1 �|vΣ � 5 Let α ≡ �Λ = 1�AL+ �Λ = 0� β ≡ �Λ = 1�L+ �Λ = 0� and use �1|H|2� = (−1)2J+S 1+S 2+σ 1+σ 2�−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. (10 points) Construct the e/f parity basis using the following phase definitions: σv |v, nΛσ SΣ, 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. (5 points) Factor the 9 × 9 Hamiltonian into a 5 × 5 and a 4 × 4 matrix using the e/f basis functions. D. Obtain distortion correction terms for the �(53 points) � � the centrifugal � � � Π0 |H|3 Π0 e , 3 Π1 |H|3 Π1 e , and 3 Π1 |H|3 Π0 e matrix elements. DΠ ≡ − � �vΠ |B |v � �2 Π 0 − E0 EΠ Πv � � v vΠ E. 3 + (5 points) Obtain the correction terms for the effect of remote � � � � Σ levels on 3 Π for the 3 Π0 |H|3 Π0 e and f and 3 Π1 |H|3 Π1 e and f matrix elements. o≡ � [(1/2)α �vΠ |v � �]2 Σ 0 − E0 E Πv Σv � � vΣ p≡4 � [(1/2)αβ �vΠ |v � � �vΠ |B |v � �] Σ Σ 0 − E0 EΠ Σv � � v vΣ � [β �vΠ |B|v � �]2 Σ q≡2 0 − E0 EΠ Σv � � v vΣ Express diagonal contributions to the Λ–doubling of the Ω = 0 and 1 3 Π substates in terms of o, p, and q. 6