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573lec31

# 573lec31 - 5.73 Lecture#31 31 1 MATRIX ELEMENTS OF F(i AND...

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31 - 1 5.73 Lecture #31 updated September 19, verify that (N!) –1/2 is correct factor MATRIX ELEMENTS OF F(i) AND G(i,j) Last time: orbitals configurations states (“terms”) Fermions: Slater Determinants: Pauli Exclusion Principle TODAY: 1. SLATER DETERMINANTAL MATRIX ELEMENTS Recall: specify standard order (because Determinant changes sign upon binary permutation) Goal: make inconvenience of Slater determinants almost vanish — matrix elements will be almost what you expect for simple non-antisymmetrized products of spin-orbitals. pages 31-2,3,4 are repeat of 30-6, 7,8 A. Normalization: ψ N p N N u u N = ( ) [ ] ! ( ) ( ) ( ) / 1 2 1 1 1 * * * u u k u k u u i i i i j j i are othonormal u(i) u(j) has no meaning because bra and ket must be associated with SAME e only nonzero LEGAL terms in are those where EACH otherwise get AT LEAST 2 MISMATCHED bra - kets ℘℘ , ( ) ( ) ( ) ( ) = = = P P 0 0 A. Normalization B. (i) One - e operator e.g. C. (i, j) Two - e operator e.g. e F H s G H r SO i i i i ij i j a r e = ( ) = > 2 Notation for Slater Determinant: main diagonal . ψ ψ N N p p N N p p i N i i i i N u u N u u N re N u u = [ ] [ ] = + + = ( !) ( ) ( ) ( ) ( ) ( ) ( !) ( ) , , 1 1 1 1 1 1 1 1 1 ℘ ℘ ℘ ℘ arrange into products of one - e overlap integrals: - P P (Here the electron names match in each bra-ket but the spin-orbitals do not match.) Think of a one- or two-e operator as a scheme for dealing with or “hiding” the small number of mismatched spin-orbitals.

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31 - 2 5.73 Lecture #31 updated September 19, B. Matrix elements of one-electron operators Product of N orbital matrix element factors in each term of sum. Of these, N–1 are orbital overlap integrals and only one involves the one-e operator. Thus it is necessary that and = = = + = ( ) [ ] = = + , , ( ) ! ( ) ( ) ( ) ( ) p p N u u u N u N p p N N N N 1 1 1 1 1 1 1 1 1 ψ ψ each term in sum over gives + 1, but there are N possibilities for possibilities for 1 2 P P , N 1 = ( ) = N N N N ! ! possibilities for sum over ℘℘ ψ ψ 1 1 1 ψ ψ A p N B p N N a a N N b b N ( ) ( ) ( ) ( ) ! ( ) ( ) ! ( ) ( ) / / 1 2 1 1 2 1 1 1 1 1 ψ ψ A B i p p i i p p i i i i i N a f b N a b a i f b i F r P P P r P = ( ) ( ) [ ] ( ) [ ] = ( ) ( ) [ ] ( ) + + ! ( ) ( ) ! ( ) ( ) ( ) ( ) , , , , 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ℘℘ ℘℘ P P N N N N a N b N ( ) ( ) [ ] F r L = ( )
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573lec31 - 5.73 Lecture#31 31 1 MATRIX ELEMENTS OF F(i AND...

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