Employing a direct calculation of the TDC is more efficient than calculating the NAC, and it can be shown to yield more numerically accurate integration of the TSH electronic equations of motion than direct use of the NAC, especially for situations of trivial crossings. More complete semiclassical methods, like semiclassical Ehrenfest (SE) dynamics and coherent switching with decay of mixing (CSDM) dynamics, require all components of the NAC. Because the evaluation of the time derivative is more efficient than the evaluation of a NAC, we have recently formulated an effective NAC that can be computed from the TDC and that can be used to carry out SE and CSDM dynamics with only the time derivative (without requiring a full NAC calculation). We also reported test calculations showing that using the effective NAC instead of the actual NAC does not cause a large error. These variants of SE and CSDM are called tSE and tCSDM, where "t" denotes time derivative. The TDC needed for such calculations can be approximated in various ways by overlaps of electronic wave functions at the current and previous time step of the trajectory. Therefore, TSH, tSE, and tCSDM can be all be calculated efficiently without computing NACs from electronic structure software.