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Significant Conditions on the Two-electron Reduced Density Matrix from the Constructive Solution of N-representability

We recently presented a constructive solution to the N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2,q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2,q)-positivity conditions to higher (2,q+1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2,q)-positivity conditions for q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.