Generalized Pauli Constraints in Reduced Density Matrix Functional Theory

The ground state of a system is determined within this approach by minimizing the energy functional with respect to the 1RDM while satisfying that the 1RDM corresponds to a fermionic ensemble (Coleman’s conditions). As the explicit expression of the energy functional with respect to the 1RDM is not known, different approximate functionals are employed. If we had the exact functional performing the energy minimization using the ensemble representability constraints would be enough to find a 1RDM that corresponds to a pure state (if our ground state is not degenerate so we really have a pure state). However, performing the energy minimization with approximate functionals, as we found for 3 electron systems test cases, results in occupation numbers that do not satisfy the generalized Pauli constraints (GPC). One then could in principle employ the GPC as additional constraints during the energy minimization to ensure that the ground state 1RDM that finds can result from a pure state. However due to the big number of these constraints this is not feasible in practice apart from a few cases of really small systems.
An idea to be explored is constructing energy functionals that satisfy at least some of the GPC. This could serve as a change of paradigm for functional derivation because until now 1RDM functionals were mostly tested on whether they reproduced or not ground state energies correctly. Another idea that we would like to discuss is whether we could apply the GPC in an approximate way to only the electrons that have occupations smaller than one. The electrons with occupation one do not play any role to whether the 1RDM corresponds to a pure state or not. Although it is doubtful whether 1RDMs with occupations that are exactly one can correspond to a ground state of a real fermionic system, in many cases this is a sensible approximation that would significantly reduce the amount of GPC to be considered in a RDMFT minimization.