# Electron interactions between different positions or different orbitals

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?

The non-interacting orbitals in exact KS-DFT still interact between different distances and different orbitals, so the answer is (1) and (2).

Your spatial representation is basically a projection into position eigenstates, while the real orbital has contributions from all over the system. To evaluate the one-orbital contributions correctly, you need to integrate over the system, giving you (1).

Next, already the Coulomb contribution, which is included in the exact Kohn-Sham functional, includes interactions between orbitals (between orbital contributions to the total density) which gives (2).

The "non-interacting" just means that the Kohn-Sham wave function is a single Slater determinant, while the true wave function consists of exponentially many determinants.

Addendum: what you attempted to do with the grid, i.e. a real-space solution of the wave function can actually be achieved with e.g. finite differences or finite elements, see my open access review in Int J Quantum Chem 119, e25968 (2019).

• +1 for another very timely and helpful answer indeed! And congratulations on reaching 12000 points here :) Also, on reaching 400/400 score on the DFT tag! Aug 25, 2021 at 19:12
• Thank you very much for the answer. I understood the meaning of the "non-interacting" in DFT.
– neco
Aug 26, 2021 at 0:07