A Simple One-Electron Expression for Electron Rotational Factors

Within the context of FSSH dynamics, one often wishes to remove the angular component of the derivative coupling between states $\left|J\right>$ and $\left|K\right>$. In a set of previous papers, Truhlar {\em et al.} posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a {\em one-electron operator} $\hat{O}$ whose matrix element $\left<J|\hat{O}|K\right>$ was the angular component of the derivative coupling. Here, we show that a one-electron operator can in fact be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and be of immediate use for FSSH calculations.