The first new condition for the integration box under consideration is the linear condition. This condition tests whether the current TD-ADGA integral is well approximated by assuming that the PES is well described by a multi-linear interpolation between the SPs that constitute the integration box boundaries. For the one-dimensional grids, this multi-linear interpolation of the PES is simply a linear interpolation between the two grid points constituting the box boundaries. For a two-mode grid with m = (m 1 , m 2 ) and a two-dimensional box consisting of the four grid points {(q m 1 ,1 , q m 2 ,1 ), (q m 1 ,2 , q m 2 ,1 ), (q m 1 ,1 , q m 2 ,2 ), (q m 1 ,2 , q m 2 ,2 )}, the multi-linear interpolation function inside the integration box is