Ehrenfest dynamics with spontaneous localization

We propose Ehrenfest Dynamics with Spontaneous Localization (SLED), a decoherence-corrected extension of Ehrenfest dynamics based on the Gisin-Percival quantum-state diffusion equation. In SLED, the electronic wave function evolves stochastically in the adiabatic energy basis, producing trajectory-level localization. The trajectory ensemble reproduces a Lindblad-type propagation of the reduced electronic density matrix. This approach ensures linearity, trace preservation, and complete positivity, providing a physically consistent alternative to ad hoc decoherence corrections commonly adopted in mixed quantum-classical methods. Benchmark simulations on one-dimensional Tully models and multidimensional spin-boson Hamiltonians demonstrate that SLED reproduces electronic populations and captures the essential features of coherence decay. The tests, however, also reveal that accurate treatment will require generalizing the localization kernel controlling the electron-nucleus coupling strength from a constant into a function of time and phase space coordinates. SLED is implemented in the newly developed Skitten program and will be integrated into Newton-X. While the present study serves as a proof of concept, SLED establishes a rigorous and extensible framework that bridges mixed quantum-classical dynamics with open quantum system theory.