Open-shell molecular electronic states from the parametric two-electron reduced-density-matrix method.

The parametric variational two-electron reduced-density-matrix (2-RDM) method, developed from an analysis of positivity (N-representability) constraints on the 2-RDM, is extended to treat both closed- and open-shell molecules in singlet, doublet, and triplet spin states. The parametric 2-RDM method can be viewed as using N-representability conditions to modify the 2-RDM from a configuration interaction singles-doubles wave function to make the energy size extensive while keeping the 2-RDM approximately N-representable [J. Kollmar, Chem. Phys. 125, 084108 (2006); A. E. DePrince and D. A. Mazziotti, Phys. Rev. A 76, 049903 (2007)]. Vertical excitation energies between triplet and singlet states are computed in a polarized valence triple-zeta basis set. In comparison to traditional single-reference wave function methods, the parametric 2-RDM method recovers a larger percentage of the multireference correlation in the singlet excited states, which improves the accuracy of the vertical excitation energies. Furthermore, we show that molecular geometry optimization within the parametric 2-RDM method can be efficiently performed through a Hellmann-Feynman-like relation for the energy gradient with respect to nuclear coordinates. Both the open-shell extension and the energy-gradient relation are applied to computing relative energies and barrier heights for the isomerization reaction HCN(+)<-->HNC(+). The computed 2-RDMs very nearly satisfy well known, necessary N-representability conditions.