Unifying Decoherence and Phase Evolution in Mixed Quantum-Classical Dynamics through Exact Factorization

We propose mixed quantum-classical equations of motion that unify electronic coherence and phase evolution simultaneously within the exact factorization framework. Our derivation shows that incorporating the second-order electron-nuclear correlation terms from the exact coupled time-dependent Schr\"odinger equations is essential to recover both correct phase dynamics and complete electronic (de)coherence, including their effect on nuclear forces. Benchmark calculations on one- and two-dimensional model systems confirm that the approach accurately captures key nonadiabatic features. The equations therefore provide a rigorous first-principles foundation for mixed quantum-classical description of coupled electron-nuclear dynamics, bringing electronic coherence and phase evolution-long treated through separate heuristic corrections-into a single unified and systematically derived framework.