Solving for many-body stationary states using the Geminal Density Matrix

The ground state energy of an arbitrary many-electron system can be calculated by minimizing its total energy as a functional of the two-body reduced density matrix (2-RDM). Such a minimization yields exact results only if the 2-RDM is sufficiently constrained so that it corresponds to a valid many-body wave function. Despite significant progress over the last few decades, a complete set of these so-called N-representability constraints has not been determined. This work approaches the problem by expanding the 2-RDM in a complete set of two-electron eigenstates and analyzing the matrix formed by the expansion coefficients. This matrix, which we call the geminal density matrix (GDM), is found to evolve unitarily in time by the Liouville-Von Neumann equation. By studying the time evolution induced by a Hamiltonian that slowly switches on the Coulomb interaction, we show by the adiabatic theorem that matrices representing eigenstates of a non-interacting system can be evolved into those for a system with electron-electron interactions in a manner which preserves N-representability.