Coupled nuclear and electronic ground-state motion from variational reduced-density-matrix theory with applications to molecules with floppy or resonant hydrogens

The variational two-electron reduced-density-matrix (2RDM) method for electronic systems [ Phys. Rev. Lett. 93 213001 (2004)] is extended to compute ground-state distributions of electrons and hydrogen nuclei in molecules beyond the Born-Oppenheimer approximation. While traditional methods for nuclei rely on the construction of expensive potential energy surfaces or other approximations, the variational 2RDM method has the advantage of treating both electrons and hydrogen nuclei as quantum-mechanical particles simultaneously. Because these particles interact by pairwise Coulombic potentials, the ground-state energy is expressible as a linear functional of three 2RDMs corresponding to two electrons, two hydrogens, and one electron and one hydrogen. Nuclei other than hydrogen are treated in the Born-Oppenheimer approximation. Variational optimization of the ground-state energy requires that the 2RDMs be restricted by N-representability conditions to represent a realistic N-particle system where N is the total number of electrons and hydrogens. Recent progress in electronic systems with (i) developing necessary N-representability conditions and (ii) optimizing the ground-state energy subject to these conditions is extended to systems with two types of particles, electrons and nuclei. The nuclear-electronic 2RDM method can be applied to studying macroscopic quantum phenomena in molecules with “floppy” or resonant hydrogens. Illustrative applications are made to (i) large-scale hydrogen motion in hydrogen-bonded molecules and protonated acetylene C₂H₃⁺ and (ii) hydrogen resonance in malonaldehyde C₃H₄O₂ and ammonia NH₃.