Fewest-switches surface hopping and decoherence in multiple dimensions

In a recent article (Subotnik, J. E.; Shenvi, N. J. Chem. Phys.2011, 134, 24105), we introduced a new approach for incorporating decoherence into the fewest-switches surface-hopping (FSSH) algorithm, titled augmented FSSH (A-FSSH). The A-FSSH algorithm was designed to correct the well-known overcoherence problem in traditional FSSH, and thus allow wave packets on different surfaces to separate naturally subject to different forces. As presented earlier, however, the A-FSSH algorithm was restricted to two electronic states. We now extend the method to more than two electronic states and present several new model problems with multiple electronic and nuclear dimensions. Lastly, starting with the quantum Liouville equation, we rederive and implement the new phase correction suggested by Shenvi (Shenvi, N.; Subotnik, J. E.; Yang, W. J. Chem. Phys.2011, 135, 24101) and co-workers for propagating the electronic amplitude along a specified nuclear trajectory and find much improved results in certain cases.