Molecular Spectra and Molecular Structure - 1976 Ex1

Examination #1 Closed Book Slide Rules and Calculators Permitted Answer any THREE of the four questions. You may work a fourth problem for extra credit. All work will be graded but no total grade will exceed 80 points. 1. A. (10 points) Give a concise statement of Hund’s three rules. B. (10 points) State the definition of a vector operator. C. (10 points) If B and C are vector operators with respect to A, then what do you know about matrix elements of B·C in the |AMA � basis? D. ( 5 points) The atomic spin-orbit Hamiltonian has the form � ξ(ri )�i · si HSO = i Classify HSO as vector or scalar with respect to J, L, and S. State whether HSO is diagonal in the |JM J LS � or |LML S MS � basis. 2. Consider the following multiplet transition array: J ? Upper ? State Lower State (L�� , S�� ) ? ? (180) (16) 16934.63 48.15 16982.78 64.20 ? (0) 17046.98 50.51 50.48 (240) 17033.29 (16) 17097.46 64.17 63.10 (L� ,S� ) ? (310) 17160.56 Intensities are in parentheses above transition frequencies in cm−1 ; line separations in cm−1 are given between relevant transition frequencies. page 2 A. (10 points) Use the Landé interval rule E(L, S , J) − E(L, S , J − 1) = ζ(nLS )J to determine J� and J�� values. Rather than list the J� and J�� assignments of each line, only list J� and J�� for the line observed to be most intense and for the line observed to be least intense. B. (10 points) Use the range of J� and J�� and the intensity distribution (i.e., that the most intense transition is not ΔJ = 0) to determine the term symbols (2S +1 L) for the upper and lower states. Assume ΔS = 0. C. ( 5 points) Is the upper state regular (highest J at highest term energy) or inverted (highest J at lowest term energy)? Is the lower state regular or inverted? [Partial energy level diagrams might be helpful here.] 3. A. ( 5 points) List the L-S terms that arise from the (ns)(np)2 and (ns)2 (np) configurations. [HINT: (np)2 gives 1 S, 3 P, 1 D; to get sp2 couple an s electron to these three states.] B. ( 5 points) Which configuration gives rise to odd terms and which to even? C. ( 5 points) List the electric dipole allowed transitions between terms of the sp2 and s2 p configurations. (Ignore fine-structure splitting of L-S terms into J-states.) D. (10 points) Construct qualitative energy level diagrams on which you display all allowed J�� –J� compo­ nents of 2 P◦ − 2 S , 2 P◦ − 2 P, and 2 P◦ − 2 D transitions. Indicate which J�� –J� line you would expect to be strongest for each of these three transitions. 4. (25 points) Calculate transition probabilities for the two transitions nsnp 1 P◦10 → (np)2 1 S 00 nsnp 1 P◦10 → (np)2 1 D20 given the following information: 1 ◦ P10 1 1 S 00 D20 = |J = 1, M J = 0, L = 1, S = 0� 1 1 = √ |s0− p0+ | − √ |s0+ p0− | 2 2 ≡ |J = 0, M J = 0, L = 0, S = 0� 1 1 = √ |p1− p − 1+ | − √ |p1+ p − 1− | + 3 3 1 1 ≡ √ |p1+ p − 1− | − √ |p1− p − 1+ | + 6 6 1 √ |p0+ p0− | 3 2 √ |p0+ p0− | 6 The electric dipole transition moment operator, µ, does not operate on spin coordinates, is a one-electron operator, and is a vector with respect to �i . nsnp → (np)2 transitions are Δ� = +1 processes. The relevant page 3 Δ� = +1 matrix elements, as given by the Wigner-Eckart theorem for vector operators are � � �� �� 1 1 � � n, � = 1, m� = 1 �� (µ+ + µ− )�� n, � = 0, m� = 0 = − √ µ+ (ns) 2 2 � � �� �� n, � = 1, m� = 0 �µz � n, � = 0, m� = 0 = µ+ (ns) � � �� �� 1 1 � � n, � = 1, m� = −1 �� (µ+ + µ− )�� n, � = 0, m� = 0 = + √ µ+ (ns) 2 2 where µ+ (ns) is the reduced matrix element �np�µ�ns�. Show all your work including false starts. If you are unable to express the transition probabilities in terms of µ+ (ns), lavish partial credit will be given for the ratio of transition probabilities 1 P◦ 10 1 P◦ 10 − 1 S 00 − 1 D00 .