The systems we will limit ourselves to, are all systems which can be described by a restricted basis set, We will thus factorize the full determinant $\vert\hat{D}\vert$ into two smaller ones, where each can be identified with $\uparrow$ and $\downarrow$ respectively:

$$\vert\hat{D}\vert = \vert\hat{D}\vert_\uparrow\cdot \vert\hat{D}\vert_\downarrow$$