When I consider the grid positions $r$ in a molecular system, I have a matrix, where each row can be regarded as an N-dimensional vector and each column can be regarded as a G-dimensional vector. In the following figure, N is the number of orbitals and G is the number of grid positions.
In this case, the elements of $i$th N-dimensional vector is the values of orbital functions on the position $r_i$; the elements of $j$th G-dimensional vector is the values of $j$th orbital function.
In this matrix, I believe that there are two kinds of interactions. (1) The interactions between different distances (red arrow). (2) The interactions between different orbitals (blue arrow) .
The DFT concept is based on the non-interacting electrons moving in the effective potential. Here, is the meaning of non-interacting electrons (1) or (2) or both?