The Normal-Ordered Electronic Hamiltonian

The contraction rules we examined earlier (cf. Eqs. [92] and [93]) state that since the creation operator is on the left, the contraction is zero unless af, and a both act in the hole space and give 8. This simplifies the one-electron part of the equation to 

Now we rewrite the string of annihilation and creation operators from the two-electron part of H as 

all these contractions are zero unless the leftmost operator of the contraction acts in the hole space. This leads to the simplified form 

Remembering that for antisymmetrized two-electron integrals in Dirac s notation, pqWrs) = - pqWsr) = - ( pllrs) = qpWsr), we may re-index sums and combine terms where appropriate to obtain 

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write 

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