Therefore e2 y1 s 1 s 1 s 1 s 0 r12 hw6nbpoint we

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Unformatted text preview: affected by the operator, and the matrix element is proportional to X2 s » 1 s \ = 0. The same is true also with the second term where the orthogonality of the second electron state makes it vanish. The only contribution comes from the third term. By going through the same steps as for the diagonal piece, we find - X1 sÆ 1 s 2 s » D H » 1 sÆ 2 s 1 s \ 2 2 2 e e e = - Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ … 1 sÆ 2 s 1 s ] r12 r13 r23 2 2 2 e = - Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 2 s 1 s ] r12 where the "2" and "3" are relabeled to "1" and "2". Out of 6 ä 5 = 30 (or in general N ! ä HN ! - 1L) off-diagonal matrix ele1 ments, there are only 6 ä 3 = 18 (or in general N ! äN C2 = N ! N HN - 1L ê 2) such terms. The overall N ! cancels ÅÅÅÅÅ!ÅÅ in the N normalization factor. 2 On the other hand, when the ket and bra has more than two electrons interchanged, the orthogonality of the single-particle states makes them vanish. For exam ple, - X1 sÆ 1 s 2 s » D H » 1 sÆ 2 s 1 s \ e e e = - Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ … 1 sÆ 2 s 1 s ] r12 r13 r23 2 2 2 e e e = - Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅ … 1 sÆ 2 s 1 s ] - Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 2 s 1 s ] - Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅ … 1 sÆ 2 s 1 s ] r12 r13 r23 In each term, there is one electron that is not affected by the operator that makes the matrix element vanish. 2 2 2 Therefore, the off-diagonal pieces contribute as e2 e2 e2 - Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 sÆ ] - Y1 sÆ 2 s … ÅÅÅÅÅÅÅ … 2 s 1 sÆ ] - Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 2 s 1 s ]. r 12 r12 r12 The grand total is X1 s2 2 s » D H » 1 s2 2 s\ e e e = Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s ] + Y1 sÆ 2 s … ÅÅÅÅÅÅÅ … 1 sÆ 2 s] + Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 s 2 s] r12 r12 r12 2 2 2 e e e - Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 sÆ ] - Y1 sÆ 2 s … ÅÅÅÅÅÅÅ … 2 s 1 sÆ ] - Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 2 s 1 s ] r 12 r12 r12 2 2...
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