From Populations to Coherences: A First-Principles Framework for (De)coherence in Mixed Quantum–Classical Dynamics

Nonadiabatic dynamics methods often rely on artificial decoherence schemes to correct the decay of electronic coherences driven by nuclear motion of state populations. Working in the Wigner–Moyal representation, we derive an exact relation expressing the electronic coherence as a non-commutative product of population distributions and a dynamical phase. Substitution of this identity into the quantum Liouville equation yields new equations of motion and establishes an explicit closure relation between populations and coherences in phase space. This closure provides a rigorous link between population and coherence dynamics and enables reconstruction of quantum interference directly from trajectory-ensembles without empirical corrections. We demonstrate this framework for dynamics generated by a vertical excitation in a displaced Morse oscillator model. Our results reproduce quantum-interference features lost in semiclassical and mixed quantum classical approximations and clarify the microscopic origin of decoherence as emerging intrinsically from population dynamics. This formulation offers a first-principles approach for developing quantum-consistent, mixed quantum–classical theories and provides a microscopic basis for transition probabilities in trajectory-based nonadiabatic simulations.