Time-dependent nuclear-electronic orbital Hartree-Fock theory in a strong uniform magnetic field

In an ultrastrong magnetic field, with field strength B ≈ B₀ = 2.35 × 10⁵ T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born-Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and processes may play a more important role in this mixed-field regime than in the weak-field regime on Earth. To understand the chemistry in the mixed regime, it therefore becomes important to explore non-BO methods. In this work, the nuclear-electronic orbital (NEO) method is employed to study protonic vibrational excitation energies in the presence of a strong magnetic field. The NEO generalized Hartree-Fock theory and time-dependent Hartree-Fock (TDHF) theory are derived and implemented, accounting for all terms that result as a consequence of the nonperturbative treatment of molecular systems in a magnetic field. The NEO results for HCN and FHF^- with clamped heavy nuclei are compared against the quadratic eigenvalue problem. Each molecule has three semi-classical modes owing to the hydrogen-two precession modes that are degenerate in the absence of a field and one stretching mode. The NEO-TDHF model is found to perform well; in particular, it automatically captures the screening effects of the electrons on the nuclei, which are quantified through the difference in energy of the precession modes.