Cancels in the n normalization factor 2 e e e y1 s 1

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Unformatted text preview: 1 sÆ 2 s 1 s \ + X2 s 1 s 1 sÆ » D H » 2 s 1 s 1 sÆ \ + X1 s 1 sÆ 2 s » D H » 1 s 1 sÆ 2 s \L = X1 sÆ 1 s 2 s » D H » 1 sÆ 1 s 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 1 In general, there are N ! diagonal matrix elements which cancels the ÅÅÅÅÅ!ÅÅ normalization factor, and each term contributes N C2 N pieces, one for each Coulomb term. 2 2 2 Unlike for the single-body operator, there are also the so-called "exchange terms" where two electrons are interchanged in the ket and the bra, - 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 the first term, the third electron is not 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 2 2 2 In general, there are N ! diagonal matrix elements which cancels the ÅNÅ!ÅÅ normalization factor, and each term contributes N C2 ÅÅÅ pieces, one for each Coulomb term. HW6.nbfor the single-body operator, there are also the so-called "exchange terms" where two electrons are interchanged in the4 Unlike ket and the bra, - 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 the first term, the third electron is not...
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