Nakatsuji's theorem of the necessary and sufficient conditions of the wave function revisited

We will here revisit Nakatsuji's theorem of the necessary and sufficient conditions of the wave function and reinterpret these conditions in the new light of our findings. It will here be shown that the equations for the necessary and sufficient conditions are not independent and that these equations can be reduced to a single equation. This observation reduces the number of conditions for the wave function for the electronic Hamiltonian to (Formula presented.), where (Formula presented.) is the number of basis functions, which coincide with the number of parameters in the two-particle reduced density matrix. Since only the highest order electron interaction term determines the necessary and sufficient conditions Nakatsuji's theorem can in this way be interpreted as a generalized Brillouin theorem. In this way the stationary conditions for the wave function of Hamiltonians with any n-body interaction takes a similar form. It is hoped that this new interpretation of the necessary and sufficient conditions as a generalized Brillouin theorem can give insights into the development of novel and compact representations of the wave function.