The proper electron interaction hamiltonian is
This hamiltonian is correct because it gives the correct expectation values. The average value with respect to a two-electron state is kl\Hint\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 uk and ul in “first-quantization”. The full electronic hamiltonian can be expressed as
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, ...