In the Moller-Plesset Perturbation Theory, the second order correction to the ground state energy can be divided into a same-spin and an opposite-spin terms by integrating the spin out. This allows to get integrals using only space orbitals, just as specified on the psi4 website here: https://psicode.org/psi4manual/master/dfmp2.html
I was able to get the expresion for opposite spins. However, when trying to perform the integration for same spins, I haven't been able to get rid of one of the terms and get the expresion I want. This happens because all four orbitals have the same spin function, so every term should mantain after the integration.
Something’s obviously missing here. The terms that describe the contribution of ``opposite spins" can come from, either the sector of total spin-1, or the sector of total spin-0. The terms that describe the contribution of ``same spins" can only come from the sector of total spin-1. So the statement that the second order correction can be divided in the way described is incomplete, because it doesn’t take into account the total spin. Two spin-1/2 combine to a spin-1 sector and a spin-0 sector.
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