Diagonal matrix elements which cancels the

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Unformatted text preview: 1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s 2 s] + Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅ … 1 sÆ 1 s 2 s] + Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s 2 s] r12 r13 r23 In the first term, the third electrons are not affected by the operator and e2 Y1 sÆ 1 s 2 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 1 s 2 s] r12 2 2 2 e = JY1 sÆ …1 ⊗Y1 s …2 ⊗X2 s »3 L ÅÅÅÅÅÅÅ H » 1 sÆ \1 ⊗ … 1 s ] ⊗ … 2 s] N r12 2 = Y1 s …1 ⊗Y1 s Æ Æ = Y1 s …1 ⊗Y1 s e2 …2 ÅÅÅÅÅÅÅ r12 e2 …2 ÅÅÅÅÅÅÅ r12 … 1 s ] ⊗ … 1 s ] X2 s »3 » 2 s\3 2 Æ 3 …1s ] ⊗…1s ] 1 Æ 2 = Y1 s 1 s … ÅÅÅÅÅÅÅÅ … 1 s 1 s ] r12 Similarly with the other two terms. Therefore, X1 sÆ 1 s 2 s » D H » 1 sÆ 1 s 2 s\ = Æ e2 Æ 1 2 e e e Y1 sÆ …1 ⊗Y1 s …2 ÅÅÅÅÅÅÅ … 1 sÆ ] ⊗ … 1 s ] + Y1 sÆ …1 ⊗Y2 s …3 ÅÅÅÅÅÅÅÅ … 1 sÆ ] ⊗ … 2 s] + Y1 s …2 ⊗Y2 s …3 ÅÅÅÅÅÅÅ … 1 s ] ⊗ … 2 s] r 12 r13 r23 2 1 2 2 1 2 2 3 3 Because each term involves only two electrons, not three of them, we can relabel them so that the Coulomb potential always refers to "1" and "2", X1 sÆ 1 s 2 s » D H » 1 sÆ 1 s 2 s\ = e e e Y1 sÆ …1 ⊗Y1 s …2 ÅÅÅÅÅÅÅ … 1 sÆ ] ⊗ … 1 s ] + Y1 sÆ …1 ⊗Y2 s …2 ÅÅÅÅÅÅÅÅ … 1 sÆ ] ⊗ … 2 s] + Y1 s …1 ⊗Y2 s …2 ÅÅÅÅÅÅÅ … 1 s ] ⊗ … 2 s] r 12 r12 r12 2 2 1 2 1 2 1 2 2 and we write them in a simpler expression, e2 e2 e2 X1 sÆ 1 s 2 s » D H » 1 sÆ 1 s 2 s\ = Y1 sÆ 1 s … ÅÅÅÅÅÅÅ … 1 sÆ 1 s ] + Y1 sÆ 2 s … ÅÅÅÅÅÅÅÅ … 1 sÆ 2 s] + Y1 s 2 s … ÅÅÅÅÅÅÅ … 1 s 2 s] r12 r12 r12 Namely the sum of all three combinations. The same applies to all the diagonal pieces in the expectation value, and they all give the same result. The contribution of all diagonal terms is hence 1 ÅÅÅÅ HX1 sÆ 1 s 2 s » D H » 1 sÆ 1 s 2 s\ + X1 s 2 s 1 sÆ » D H » 1 s 2 s 1 sÆ \ + X2 s 1 sÆ 1 s » D H » 2 s 1 sÆ 1 s \ + 6 X1 sÆ 2 s 1 s » D H...
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