Orbital angular momentum constraints in the variational optimization of the two-electron reduced-density matrix

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.