Pure- N -representability conditions of two-fermion reduced density matrices
David A. Mazziotti · Physical Review A · 2016
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure $N$-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure $N$-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure $N$-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble $N$-representability conditions. Constraints on higher-order $p$-RDMs where $p>2$ are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.