Single constant localization kernel balancing population and coherence in Tully 1 (low momentum)

Determine whether there exists a single constant value of the localization kernel k used in SLED (Ehrenfest Dynamics with Spontaneous Localization based on the Gisin–Percival quantum-state diffusion equation with adiabatic energy eigenstates) that simultaneously yields accurate electronic state 1 population and accurate coherence between states 1 and 2 for the one-dimensional Tully 1 model at low initial momentum (p0 = 10 a.u.).

Background

SLED introduces decoherence into Ehrenfest dynamics via the Gisin–Percival QSD equation, using a localization operator L = kH with k treated as a constant in this proof-of-concept work. For the standard one-dimensional Tully test models, k is tuned to reproduce benchmark populations and coherences.

In the Tully 1 model at low initial momentum, the authors report that tuning a single constant k could not achieve simultaneous agreement for both population and coherence. This failure motivates the later discussion on generalizing k to depend on time and phase-space variables, but the immediate question remains whether any constant k can achieve the desired balance in this specific case.