As observed for CTMQC, the additional coupled-trajectory contribution that depends on the quantum momentum in Eq. ( ) induces decoherence effects in CTTSH as well. Instead, to achieve decoherence in a standard TSH procedure, the original algorithm needs to be modified using decoherence corrections, as we will do in Section III employing the widely used energy-based decoherence (ED) correction of Ref. [65].