Estimating the entropy and quantifying the impurity of a swarm of surface-hopping trajectories: a new perspective on decoherence

In this article, we consider the intrinsic entropy of Tully's fewest switches surface hopping (FSSH) algorithm (as estimated by the impurity of the density matrix) [J. Chem. Phys. 93, 1061 (1990)]. We show that, even for a closed system, the total impurity of a FSSH calculation increases in time (rather than stays constant). This apparent failure of the FSSH algorithm can be traced back to an incorrect, approximate treatment of the electronic coherence between wavepackets moving along different potential energy surfaces. This incorrect treatment of electronic coherence also prevents the FSSH algorithm from correctly describing wavepacket recoherences (which is a well established limitation of the FSSH method). Nevertheless, despite these limitations, the FSSH algorithm often predicts accurate observables because the electronic coherence density is modulated by a phase factor which varies rapidly in phase space and which often integrates to almost zero. Adding "decoherence" events on top of a FSSH calculation completely destroys the incorrect FSSH electronic coherence and effectively sets the Poincaré recurrence time for wavepacket recoherence to infinity; this modification usually increases FSSH accuracy (assuming there are no recoherences) while also offering long-time stability for trajectories. In practice, we show that introducing "decoherence" events does not change the total FSSH impurity significantly, but does lead to more accurate evaluations of the impurity of the electronic subsystem.