N representability condition for

For the next step, we show how we consider the N-representability conditions for the 1-RDM y for a system with N particles that is all of its eigenvalues should be between zero and one [17]. In other words, this condition is equivalent to saying that y and / — y are positive semidefinite, where / is the identity matrix. Assuming that H is the one-body Hamiltonian, we have... 

Minimization of the functional (41) has to be performed under the orthonormality requirement in Eq. (4) for the NSOs, whereas the ONs conform to the N-representability conditions for D. Bounds on the ONs are enforced by setting rii = cos y, and varying y,- without constraints. The other two conditions may easily be taken into account by the method of Lagrange multipliers. 

There are two ways to fix this problem. First, one can attempt to derive N-representability conditions for the g-density in the spatial representation. This seems hard to do, although one constraint (basically a special case of the G condition for the density matrix) of this type is known, see Eq. (77). Deriving additional constraints is a priority for future work. 

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