573lec31

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

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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 Nu u N = () −℘ [] !( ) ( ) ( ) / 12 1 11 * * * u ukuk u u i ii ij 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 ℘℘ , () () = == PP ll 00 A. Normalization B. (i) One - e operator e.g. C. (i, j) Two - e operator e.g. –e FH s GH r SO i iii ij ar e = = > l 2 Notation for Slater Determinant: main diagonal . ψψ NN pp i N ii ii u N u u N re u =− + + = ∑∏ ( ! ) ( ) ( )( ) ( ) (! ) ( ) , , 1 1 1 1 1 arrange into products of one - e overlap integrals: - (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 = = −= + = () [] == + , , ( ) !( ) ( ) ( ) ( ) pp Nu u u N u N NN N N 11 1 ψψ each term in sum over gives + 1, but there are N possibilities for possibilities for 12 PP , N 1 = = N N ! ! possibilities for sum over ℘℘ 1 ψ A p N B p N Na a N Nb b N ) ( ) ) ( ) / / 1 1 AB i i i ii i ii f b b aif bi Fr Pr P = = + + ) ( ) ) ( ) ,, 1 1 1 1 P P aN bN L = ∑∑ i i i i f e.g. = rr l Thus the assumed (N!) –1/2 normalization factor is correct.
31 - 3 5.73 Lecture #31 updated September 19, SELECTION RULE ΨΨ

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

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