The space-and time-smoothening of the one-mode density serve similar purposes as the average density of TI-ADGA. The smoothening over time and space ensures that fast oscillations and changes that might occur in the one-mode density are smoothened out, so that the TD-ADGA integral does not change wildly between iterations. The space-smoothening further expands the density to a slightly wider interval than what the wavefunction itself suggests. This forces the TD-ADGA algorithm to take into account a slightly wider interval of the PES and this may precisely be the domains where the wave function propagates within the next short time.