17_580ln_fa08

17_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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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 .
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5.80 Lecture #17 Fall, 2008 Page 1 of 9 pages Lecture #17: Hund’s Cases: 2 , 2 ± Examples 4 ways to think about Hund’s Cases: 1. Pattern forming quantum numbers. A search for 1 + -like rotational level pattern BJ(J + 1) cases a and c BN(N + 1) case b Repeated 1 -like patterns as in 3 state. o *** 2. H eff and Perturbation Theory. When is H 12 E 12 and vice versa? 3. Vector precession models. How do the various angular momenta project into the body and laboratory? What gets averaged out and what does not as vectors precess? Are various angular momenta components expected to be conserved? 4. Basis set transformations (like | JMLS 〉↔ | LM L | SM S for atoms) 3j, 6j, 9j transformation coefficients. Alternate form of H example: H ROT [ L ] 2 . = B(R) N We are going to look at the 2 , 2 + matrix and use perturbation theory to identify and describe each of the Hund’s limiting cases. Exclude γ f e 2 3/2 2 1/2 2 + y = J + 1/2 Treat 2 , 2 + together because they could form “p-complex”. 2 3/2 2 1/2 2 + E + A /2 + B (y 2 – 2) –B (y 2 – 1) 1/2 β (y 2 – 1) 1/2 sym E – A /2 + B y 2 α + β (1 y) sym sym E + B (y 2 y) E o = A 2B Crucial Energy Denominators: = 3/2 − ∏ 1/2 (spin-orbit) E ∏ − ∑ o = E E vs. p π p σ A B B A “ligand field” (also exchange splittings) for 2e configurations
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5.80 Lecture #17 Fall, 2008 Page 2 of 9 pages Crucial Coupling Terms BJ ± S ⇒− B ( y 2 1 ) 1/2 (spin-uncoupling) BJ ± L −β ( y 2 1 ) ( -uncoupling) H SO ⇒α (spin-orbit) Case (a) Strong spin-orbit, stronger non-spherical field E ∏∑ A, α By, β y , Λ , S good | n Λ S | JM patterns BJ(J + 1) one for each , separated in energy by A Λ Case (b) Weak spin-orbit, strong field E ∏∑ By, β y A, α Λ ,S good, , bad | n Λ SNJM patterns BN(N + 1) - with fine structure splittings Case (c) Strongest spin-orbit, moderate field A, α E ∏∑ By, β y good, N, S, Λ , bad | n JM (J a and a atom-in-molecule quantum numbers)
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17_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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