We will now describe how the TD-ADGA algorithm is implemented as a part of the ordinary differential equation integrator in MidasCpp. Before the time-propagation is initialized, an initial iteration of the TD-ADGA procedure is carried out to create a PES to start both the dynamics and TD-ADGA procedure from. In this zeroth iteration of the TD-ADGA algorithm, the initial PES V 0 is created using the TI-ADGA procedure using a density either provided by a VSCF calculation on some other potential or by the user on input. The placement of the initial SPs are determined by the initial density. A SP is always placed at the reference structure and two further SPs are placed such that the entire density (within the residual threshold determined by Eq. 41) is contained within the SP boundaries. A further SP is also placed at the maximum density value if this is not close to the reference structure. Using these initial SPs and the fixed initial density, a TI-ADGA calculation is carried out. This creates the grids for all MCs with initial values for all the TD-ADGA integrals and it provides a PES to the ordinary differential equation integrator which then initialize the time-dependent wave function y 0 .