The proper electron interaction hamiltonian is

with

This hamiltonian is correct because it gives the correct expectation values. The average value with respect to a two-electron state is *kl*\*H*_{int}\*kl* = (*kk\ll*) – (*kl\lk*) exactly as the result (coulomb minus exchange) of calculating the average of 1/|_{1} – _{2}| with respect to a two-electron Slater determinant with the spin orbitals *u _{k}* and

It is common to use an alternative notation for the electron-electron interaction integrals *rr*^{′}|*ss*^{′} = (*rs*|*r*^{′} *s*^{′}), *i.e.,* with the complex conjugate spin orbitals, belonging to the *a*^{†} operators, ...

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