Gauge-Symmetric Dual Lagrangian Frameworks for Born-Oppenheimer Molecular Dynamics

Born-Oppenheimer molecular dynamics delivers quantum accurate trajectories by evolving nuclei on the electronic ground-state surface while retaining electronic observables such as dipoles and polarizabilities. In Kohn-Sham density functional theory, this typically relies on an iterative self consistent field solver at each time step. Machine learning can predict electronic quantities or improve SCF initialization, yet dynamics remain sensitive to residual self consistency error, which can yield non conservative forces and energy drift, and repeated diagonalization limits differentiable and accelerator efficient implementations. We propose a gauge-symmetric dual Lagrangian framework that avoids SCF loops by propagating the physically meaningful electronic state as the occupied subspace and its density projector on a Grassmann manifold. A residual driven update stabilized by Rayleigh type dissipation with time decaying inertia yields a closed evolution of the subspace, its conjugate momentum, and the projector. Coupled with a neural gauge-symmetric Hamiltonian with analytic coordinate derivatives, the method provides closed form forces with Pulay corrections and enables stable quantum BOMD on molecular simulation benchmarks.