573lec35 - 5.73 Lecture#35 35 1 SPIN ORBIT Many-Electron(N...

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35 - 1 5.73 Lecture #35 updated September 19, death of orbital picture expansion of 1/r ij : multipoles, integrals over AOs in nucleus-centered coordinates SELECTION RULES: orbital and many-e basis sets Gaunt Coefficients: [products of 3-j] Slater-Condon Sum Rule Method - avoid necessity to derive: *eigenvectors * off diagonal elements in Slater basis Hund’s 1st and 2nd Rules abc kkk ,, FG kk () SPIN - ORBIT: Many-Electron ζ (N, L, S) Single-Orbital ζ n l Coupling Constants LAST TIME : er ij ij 2 TODAY : next time Hund’s 3rd Rule and Lande g J –factors A. General Importance of spin - orbit term operator B. replace 1 - e operator by more convenient for S = 0, L = 0 special case C. Landé Interval Rule (Patterns are for breaking! Information about "dark" states) D. matrix elements in Slater Determinantal Basis Set another operator replacement Fundamental control parameter: - scaling of * off - SO SO n H S H l l = ⋅− ⋅∆ i ii i n ar e Trick N L S Pattern n NLS l s L l 1 : , , * , ζ ζ ζζ diagonal spin diagonal spin - orbit matrix elements: L 0, S J = 0. ∆≠ ∆≠ ∆
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35 - 2 5.73 Lecture #35 updated September 19, A. Importance of spin-orbit 1. H SO produces diagnostically significant “fine structure” CONFIGURATIONAL ASSIGNMENTS L,S term assignments PATTERNS:* # components * sign of pattern (largest splitting on top or bottom) * statistical weight (2J + 1) of lowest vs. highest energy component * overall magnitude of splitting 2. for heavy atoms, H SO responsible for such large splittings and off- diagonal interactions that L-S terms “vanish”, ∆S selection rules are violated. “Inter-System Crossing (ISC)”. Need to “deperturb” to recover F k , G k parameters which should vary smoothly from atom to atom (isoelectronic series) (shielding rules) 3. spin-forbidden transitions provide energy linkages between manifolds of states with different values of S. “InterSystem Crossing (ISC)” 4. non-textbook Zeeman tuning coefficients (clues about unobserved “dark” states) Atoms, Molecules, Quantum Dots, solids: in an electronic transition, light acts on single e and operates exclusively on spatial part of ψ spin-flips are forbidden except when H SO mixes states of different S — forbidden transitions “borrow” intensity from allowed transitions. In time-domain: a short pulse prepares, at t = 0, a non-eigenstate that is a pure ∆S = 0 excitation (and ∆ l = ±1) basis state. The “pluck”!
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573lec35 - 5.73 Lecture#35 35 1 SPIN ORBIT Many-Electron(N...

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