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http://ocw.mit.edu 5.80 SmallMolecule Spectroscopy and Dynamics
Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1976 Examination #1
March 12, 1976
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 spinorbit Hamiltonian has the form
�
HSO =
ξ(ri )�i · si
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
? Lower State (L�� , S�� )
?
?
(180)
(16)
16934.63 48.15 16982.78 64.20 ?
(0)
17046.98 50.51 Upper ? State 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. 5.76 Exam #1 March 12, 1976 page 2 A. (10 points) Use the Land´ interval rule
e
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 Δ J = 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 LS 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 s p2 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 s p2 and s2 p conﬁgurations.
(Ignore ﬁnestructure splitting of LS terms into Jstates.)
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
n snp 1 P◦ → (np)2 1 S 00
10
n snp 1 P◦ → (np)2 1 D20
10
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 oneelectron
operator, and is a vector with respect to �i . n snp → (np)2 transitions are Δ� = +1 processes. The relevant 5.76 Exam #1 March 12, 1976 page 3 Δ� = +1 matrix elements, as given by the WignerEckart theorem for vector operators are
�
�
�
�
�1
� (µ + µ )� n, � = 0, m = 0 = − 1 µ (ns)
�
n, � = 1, m� = 1 �
√+
�
−�
�2 +
�
2
�
��
�µ � n, � = 0, m = 0� = µ (ns)
n, � = 1, m� = 0 � z �
�
+
�
�
�
�
�1
�
� (µ + µ )� n, � = 0, m = 0 = + 1 µ (ns)
n, � = 1, m� = −1 �
√+
�
−�
�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 . ...
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.
 Spring '04
 RobertField
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