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573lec34 - 5.73 Lecture#34 34 1 e2/rij and Slater Sum Rule...

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34 - 1 5.73 Lecture #34 updated September 19, e 2 / r ij and Slater Sum Rule Method LAST TIME: 1. L 2 , S 2 method for setting up NLM L SM S many-electron basis states in terms of linear combination of Slater determinants * M L = 0, M S = 0 block: * diagonalize S 2 (singlets and triplets) * diagonalize L 2 in same basis that diagonalizes S 2 [Recall: to get matrix elements of L 2 , first evaluate L 2 and then left multiply by 2. coupled representations nj ω s and NJLSM J 3. Projection operators: automatic projection of L 2 eigenfunctions * remove unwanted L part * preserve normalization of wanted L part * remove overlap factor L L L S S S 2 2 + + TODAY: 1. Slater Sum Rule Trick (trace invariance): MAIN IDEA OF LECTURE. 2. evaluate (tedious, but good for you) [2-e operator, spatial coordinates only, scalar wrt J,L,S ] * multipole expansion of charge distribution due to “other electrons” * matrix element selection rules for e 2 /r ij in both Slater determinantal and many-e basis sets * Gaunt Coefficients (c k ) (tabulated) and Slater-Condon (F k ,G k ) Coulomb and Exchange parameters. Because of sum rule, can evaluate mostly type matrix elements and never type matrix elements. 3. Apply Sum Rule Method 4. Hund’s 1st and 2nd Rules i j ij e r > 2 matrix elements ψ i ψ j ab ab ab ba r r ij ij 1 1 and ab cd r ij 1
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34 - 2 5.73 Lecture #34 updated September 19, 1. Slater s Sum Rule Method r r 1 r r 2 r r 12 e 1 e 2 e r e r 1 1 1 1 2 2 2 2 ( ) ( ) at at , , , , θ φ θ φ scalar with respect to J , L , S , s i but not j i , i It is almost always possible to evaluate e 2 / r ij matrix elements without solving for all | LM L SM S basis states. * trace of any Hermitian matrix, expressed in ANY representation, is the sum of the eigenvalues of that matrix (thus invariant to unitary transformation) * e r ij i j 2 / > and every scalar operator with respect to ˆ ˆ ˆ J L,S) (or has nonzero matrix elements diagonal in J and M J (or L and M L ) and independent of M J or (M L ,M S ) [W-E Theorem: J is a GENERIC ANGULAR MOMENTUM with respect to which e 2 / r ij is classified] Recall from definition of r 12 , that e 2 / r ij is a scalar operator with respect to ˆ ˆ ˆ J, L, S but not with respect to j i or i . * destroys orbital approximation $$ for electronic structure calculations * “correlation energy,” “shielding” Interelectronic Repulsion: e 2 r ij i j > r r r r r r r r r r r r r r r r r r r r 12 2 1 12 2 1 2 1 2 2 2 12 1 2 2 2 1 2 1 2 1 2 2 2 = = + = + ( ) [ ] cos , /
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34 - 3 5.73 Lecture #34 updated September 19, expand as power series in where is smaller of integrals evaluated in 2 regions : r r r r r r r r r r 12 1 1 2 1 2 2 1 < > < < < ( ) , , 1 4 2 1 0 1 r n r r Y Y ij n m n n n n n m i i n m j j = π + ( ) ( ) [ ] = =− < > + θ φ θ φ , , * not principal q.n.!
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