Fig. 1.

(Left) First-principles stopping power calculations involve time evolving a projectile (red) passing through a target medium (blue) while monitoring observables related to energy transfer between them. The initial velocity, $v_{\mathrm{proj}}$, is chosen to mitigate trajectory sampling and finite-size error using techniques from ref. 71. The coupled electron-projectile dynamics are time evolved subject to this initial condition and the work or average force on the projectile is calculated throughout the trajectory. (Right) The stopping power is related to the slope of the work that the target does on the projectile as a function of its displacement from its original position (solid). A moving average for this slope (dashed) illustrates the rate at which the stopping power estimate converges. Close collisions involve large impulses in the work that are essential to capture on average. However, if these relatively rare events are included in the sample, they can dominate the variance for sample-efficient estimates.