The gas phase methanol molecule was used as an example to compare the performance of three different MD sampling schemes and demonstrate the strength of the direct NVE scheme. For the scheme comparison, we only performed CNEO-MD since the results from conventional AIMD will be highly similar. To further show the power of the new sampling scheme along with CNEO-MD, as well as the importance of vibrational mode coupling effects, we chose three highly challenging water clusters, including the neutral water dimer(H 4 O 2 ), the protonated water trimer (H 7 O + 3 ), and the protonated water tetramer (H 9 O + 4 ) for study.