Separation of spin and the fock coordinate wave function method in the $N$-representability problem

For the wave functions of a spin multiplet we carry out, in general form, separation of the spin variables in the two-particle density matrices, which are constructed from the three spatial two-particle functions and irreducible tensor spin operators. We find nontrivial integral relations for these spatial functions, giving necessary conditions of $N$-representability. We show that these spatial components are density matrices of the Schrödinger wave function for two groups of electrons. The condition of cyclic symmetry for this function is ensured by one of the obtained integral relations.