573lec34

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

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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 l 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 LL L SS 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 ij ij er > 2 matrix elements ψ i ψ j ab ab ab ba rr ij ij 11 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 er 11 1 1 22 2 2 () at at ,, θφ scalar with respect to J , L , S , s i but not j i , l 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) * ij ij 2 / > and every scalar operator with respect to ˆˆ ˆ JL , 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 J, L, S but not with respect to j i or l i . * destroys orbital approximation $$ for electronic structure calculations * “correlation energy,” “shielding” Interelectronic Repulsion: e 2 r ij > rr r r r r r r r r r r 12 2 1 12 2 1 2 2 2 12 1 2 2 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 rr r 12 1 12 21 < > < << () , , 14 0 1 rn r r YY ij nm n n n n n m ii n m jj = π + [] = =− < > + ∑∑ θφ ,, * not principal q.n.!
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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

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