The random phase ϕ ∈ [-π, π] is sampled at the beginning of each trajectory and is keep constant throughout the evolution of that trajectory. The purpose of this phase, in the original context of the Ehrenfest+R development, is to add randomness to the energy of the classical electromagnetic field that couples to the molecules since the Ehrenfest+R method increases the energy of the classical electromagnetic field proportional to k R at each time step due to energy conservation. While this is a necessary feature to accurately describe the classical electromagnetic field, it only affects the quantum dynamics by increasing the variance between the trajectories. This decay rate k R can be time averaged to yield k R as