Nb e 5e hence y1 s 1 s 1 s 1 s r12 8a 2 2 6 e e r

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Unformatted text preview: 2 In general, the result for the two-body operators is given by the N C2 non-exhcnage and the negative signs. N C2 exchange terms, the latter with Everthing is the same for 1 s2 2 p configuration after changing 2 s to 2 p. (d) Because the Coulomb potential does not affect the spins, the electron "1" in the ket and the bra must have the same spin to give a non-vanishing contribution, and the same for the electron "2". Therefore, e2 Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 sÆ ] = 0 r12 At this point, we have to decide if 2 s electron is spin up or down. If it is spin up, e2 Y1 s 2 sÆ … ÅÅÅÅÅÅÅÅ … 2 sÆ 1 s ] = 0. r12 Therefore, the grand total is simplified to X1 s2 2 s » D H » 1 s2 2 s\ e e = Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s ] + Y1 sÆ 2 sÆ … ÅÅÅÅÅÅÅÅ … 1 sÆ 2 sÆ ] r12 r12 2 2 e e + Y1 s 2 sÆ … ÅÅÅÅÅÅÅÅ … 1 s 2 sÆ ] - Y1 sÆ 2 sÆ … ÅÅÅÅÅÅÅÅ … 2 sÆ 1 sÆ ] r 12 r12 Furthermore, the second and third terms here are the same. After using up all spin degrees of freedom, and expression simplifies to X1 s2 2 s » D H » 1 s2 2 s\ 2 2 e e e = Y1 s 1 s … ÅÅÅÅÅÅÅ … 1 s 1 s] + 2 Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 s 2 s] - Y1 s 2 s … ÅÅÅÅÅÅÅÅ … 2 s 1 s] r12 r12 r12 2 2 2 give a non-vanishing contribution, and the same for the electron "2". Therefore, e2 Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 sÆ ] = 0 r12 HW6.nbpoint, we have to decide if 2 s electron is spin up or down. If it is spin up, At this e2 Y1 s 2 sÆ … ÅÅÅÅÅÅÅÅ … 2 sÆ 1 s ] = 0. r12 Therefore, the grand total is simplified to X1 s2 2 s » D H » 1 s2 2 s\ 5 e e = Y1 sÆ 1 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s ] + Y1 sÆ 2 sÆ … ÅÅÅÅÅÅÅÅ … 1 sÆ 2 sÆ ] r12 r12 2 2 e e + Y1 s 2 sÆ … ÅÅÅÅÅÅÅÅ … 1 s 2 sÆ ] - Y1 sÆ 2 sÆ … ÅÅÅÅÅÅÅÅ … 2 sÆ 1 sÆ ] r 12 r12 Furthermore, the second and third terms here are the same. After using up all spin degrees of freedom, and expression simplifies to X1 s2 2 s » D H » 1 s2 2 s\ 2 2 e e e = Y1 s 1 s ...
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This note was uploaded on 01/17/2014 for the course PHYSICS 221b taught by Professor Staff during the Winter '08 term at University of California, Berkeley.

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