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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 Problem Set #2
1. (a) Construct the state L = 2, S = 1, J = 1, M J = 0� from the LML S MS � basis using the ladder operator
plus orthogonality technique.
(b) Construct the states L = 2, S = 1, J = 1, M J = 0� 3 D1 L = 2, S = 2, J = 1, M J = 0� 5 D1 L = 5, S = 2, J = 3, M J = 1� 5 H3 from the L ML S MS � basis using ClebschGrodan coeﬃcients. The 3 D1 function is the same as in Part
(a) and is intended as a consistency check. 2. We know that the spinorbit Hamiltonian, HSO = AL · S , is diagonal in the L S J M J � basis but not in the
L ML S MS � basis.
(a) Construct the full nine by nine HSO matrix in the L = 1 ML S = 1 MS � basis.
(b) Construct the L = 1, S = 1, J = 2, M J = 0�
L = 1, S = 1, J = 1, M J = 0� and L = 1, S = 1, J = 0, M J = 0�
functions in the L ML S MS � basis. 3P
2 3P
1
3P
0 (c) Show that the matrix elements
�
�
�
�
�
�
L = 1, S = 1, J = 2, M J = 0 �HSO � 1, 1, 2, 0 �
� �
�
�
�
1, 1, 2, 0 �HSO � 1, 1, 1, 0 �
� �
�
�
�
1, 1, 2, 0 �HSO � 1, 1, 0, 0 �
�
�
�
�
�
1, 1, 1, 0 �HSO � 1, 1, 1, 0
�
�
�
�
�
�
1, 1, 1, 0 �HSO � 1, 1, 0, 0
�
�
�
�
�
�
1, 1, 0, 0 �HSO � 1, 1, 0, 0
�
�
expressed in terms of the L ML S MS � basis in part (b) have the values expected from L·S = 1/2 J2 − L2 − S2
evaluated in the L S J M J � basis. 3. Calculate the energies for the hydrogenic systems H and Li2+ in the following states:
2 2 P1/2 (means n = 2, s = 1/2, � = 1, j = 1/2)
2 2 P3/2
3 2 P1/2
3 2 P3/2
3 2 D3/2
3 2 D5/2
Please express “energies” in cm−1 : σ = E
−1
he cm and locate the zero of energy at n = ∞. 4. Consider the (nd)2 conﬁguration.
(a) There are 10 distinct spinorbitals associated with nd; how many Pauliallowed (nd)2 Slater determinants
can you form using two of these spinorbitals?
(b) What are the L − S states associated with the (nd)2 conﬁguration? Does the sum of their degeneracies
agree with the conﬁgurational degeneracy in part (a)?
(c) What is the lowest energy triplet state (S = 1) predicted by Hund’s rules? Does Hund’s rule predict the
lowest energy singlet state?
(d) Calculate the energies of all states (neglecting spinorbit splitting) which arise from (nd)2 in terms of the
radial energy parameters F0 , F2 , and F4 . [This is a long and diﬃcult problem. The similar (np)2 problem
is worked out in detail in Condon and Shortley, pages 191193, and in Tinkham, pages 177178. The
result for (nd)2 is also given, without explanation and in slightly diﬀerent notation, Condon and Shortley,
page 202.] What relationship between F2 and F4 is required by Hund’s rules?
5. If an atom is in a (2 p)2 3 P0 state, to which of the following states is an electric dipole transition allowed?
Explain in each case.
(a) 2 p3d 3 D2
(b) 2 s2 p 3 P1
(c) 2 s3 s 3 S 1
(d) 2 s2 p 1 P1 ...
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 Spring '04
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