  The direct variational determination of the two-electron reduced-density
matrix (2-RDM) is usually carried out under the assumption that the 2-RDM is a
real-valued quantity. However, in systems that possess orbital angular momentum
symmetry, the description of states with a well-defined, non-zero z-projection
of the orbital angular momentum requires a complex-valued 2-RDM. We consider a
semidefinite program suitable for the direct optimization of a complex-valued
2-RDM and explore the role of orbital angular momentum constraints in systems
that possess the relevant symmetries. For atomic systems, constraints on the
expectation values of the square and z-projection of the orbital angular
momentum operator allow one to optimize 2-RDMs for multiple orbital angular
momentum states. Similarly, in linear molecules, orbital angular momentum
projection constraints enable the description of multiple electronic states,
and, moreover, for states with a non-zero z-projection of the orbital angular
momentum, the use of complex-valued quantities is essential for a qualitatively
correct description of the electronic structure. For example, in the case of
molecular oxygen, we demonstrate that orbital angular momentum constraints are
necessary to recover the correct energy ordering of the lowest-energy singlet
and triplet states near the equilibrium geometry. However, care must still be
taken in the description of the dissociation limit, as the 2-RDM-based approach
is not size consistent, and the size-consistency error varies dramatically,
depending on the z-projections of the spin and orbital angular momenta.
