  The computation of strongly correlated quantum systems is challenging because
of its potentially exponential scaling in the number of electron
configurations. Variational calculation of the two-electron reduced density
matrix (2-RDM) without the many-electron wave function exploits the pairwise
nature of the electronic Coulomb interaction to compute a lower bound on the
ground-state energy with polynomial computational scaling. Recently, a
dual-cone formulation of the variational 2-RDM calculation was shown to
generate the ground-state energy, albeit not the 2-RDM, at a substantially
reduced computational cost, especially for higher $N$-representability
conditions such as the T2 constraint. Here we generalize the dual-cone
variational 2-RDM method to compute not only the ground-state energy but also
the 2-RDM. The central result is that we can compute the 2-RDM from a
generalization of the Hellmann-Feynman theorem. Specifically, we prove that in
the Lagrangian formulation of the dual-cone optimization the 2-RDM is the
Lagrange multiplier. We apply the method to computing the energies and
properties of strongly correlated electrons -- including atomic charges,
electron densities, dipole moments, and orbital occupations -- in an
illustrative hydrogen chain and the nitrogen-fixation catalyst FeMoco. The dual
variational computation of the 2-RDM with T2 or higher $N$-representability
conditions provides a polynomially scaling approach to strongly correlated
molecules and materials with significant applications in atomic and molecular
and condensed-matter chemistry and physics.
