ps2_1976

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule 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 Clebsch-Grodan coeﬃcients. The 3 D1 function is the same as in Part (a) and is intended as a consistency check. 2. We know that the spin-orbit 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 spin-orbitals associated with nd; how many Pauli-allowed (nd)2 Slater determinants can you form using two of these spin-orbitals? (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 spin-orbit 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 191-193, and in Tinkham, pages 177-178. 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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