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573lec32 - 5.73 Lecture#32 32 1 Last time Matrix elements...

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32 - 1 5.73 Lecture #32 updated September 19, Last time: Matrix elements of Slater determinantal wavefunctions Normalization: (N!) –1/2 F (i): selection rule (∆s-o ≤ 1), sign depending on order G (i,j): selection rule (∆s-o ≤ 2), two terms with opposite signs TODAY: Configuration which L-S terms? L-S basis states matrix elements Method of crossing out boxes ladders plus orthogonality Many worked out examples will not be covered in lecture. MM LS , Longer term goals: represent “electronic structure” in terms of properties of atomic orbitals 1. Configuration L,S terms 2. Correct linear combination of Slater determinants for each L,S term: several methods 3. 1/r ij matrix elements F k , G k Slater-Condon parameters, Slater sum rule trick 4. H SO * ζ (NLS) — coupling constant for each L-S term in an electronic configuration * ζ (NLS) ζ n l one spin-orbit orbital integral for entire configuration * full H SO matrix in terms of ζ n l 5. Stark, Zeeman, optical transitions 6. transition strengths There are a vastly smaller number of orbital parameters than the number of electronic states. The periodic table provides a basis for rationalization of orbital parameters (dependence on atomic number and on number of electrons.) matrix elements of , g - values r r () nrn ll + 1 KEY IDEAS: *1 /r ij destroys spin-orbital labels as good quantum numbers. * Configuration splits into widely spaced L-S-J “terms.” * is a scalar operator with respect to L, S, and J thus matrix elements are independent of M L , M S , and M J . * Configuration generates all possible M L , M S components of each L-S term. * It can’t matter which M L , M S component we use to evaluate the 1/r ij matrix elements * Method of microstates and boxes: Book-keeping which L-S states are present, organize the algebra to find eigenstates of L 2 and S 2 , basis for “sum rule” method (next lecture). 1/r ij ij >
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32 - 2 5.73 Lecture #32 updated September 19, Which L-S terms belong to (nf) 2 * shorthand notation for spin - orbitals n m / e.g. 4f3 , could suppress 4 and f ( main diagonal for Slater determinant, for simple product of spin - orbitals) * standard order (to get signs internally consistent) 3 3 2 2 - 3 - 3 is my standard order for f spin orbitals * which Slater determinants are nonzero and distinct (i.e., not identical when spin - orbitals are permuted to a different ordering)? l l l αβ α αβαβ α β + () + = = 21 7 4 s f 2 - take any 2 s-o’s and list in standard order How many nonzero and distinct Slater determinants are there for f 2 ? 14 14 13 2 spin - orbitals 2 identical electrons = 91 general n p p p n l l l l + [] + : ! ! ! 22 1 1 1 subshell : one such factor for each subshell How to generate all 91 linear combinations of Slater determinants that correspond to the 91 possible LM L SM S basis states that arise from f 2 ? Next lecture.
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573lec32 - 5.73 Lecture#32 32 1 Last time Matrix elements...

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