A mapping between effective potential and non-interacting electrons moving on the potential

Question

From this question and answer, I understood the Hohenberg-Kohn theorem and found that there is a one-to-one correspondence between the external potential $V_\text{ext}$ and the electron density $\rho$ as entire functions, not pointwise.

Now I would like to know the relationship between $\rho$ and the effective potential $V_\text{eff}$ (= $V_\text{ext}$ + $V_{H}$ + $V_{xc}$, where $V_{H}$ is the Hatree term and $V_{xc}$ is the exchange-correlation term). Is there any kind of pointwise correspondence between $V_\text{eff}$ and $\rho$ in systems? I understand that the DFT can be described as the electron cloud moving in $V_\text{eff}$, that is, the electrons do not interact with each other and are spread on $V_\text{eff}$ like the following figure of Schrödinger's and DFT's views (https://www.semanticscholar.org/paper/TOPICAL-REVIEW%3A-Designing-meaningful-density-theory-Mattsson-Schultz/9e57d4b4d5eec0f1b65fe774638b6d97a99f5f53/figure/0).

Here, is there any pointwise correspondence between $V_\text{eff}$ and $\rho$? The reason for thinking this way is that, I was wondering if there is such a mapping between the effective potential and the non-interacting electron cloud, I believe that we may be able to learn the mapping by machine learning method described in the following paper.

https://www.nature.com/articles/s41467-017-00839-3

Postscript: the word of 'point-wise' mapping means the mapping between the value of $V_\text{eff}(r)$ on a point $r$ and the value of $\rho(r)$ on $r$, not the the mapping between their entire functions, $V_\text{eff}$ and $\rho$ in the system. The (non-interacting) electrons are moving on the effective potential, so if we have or can consider a mapping between the scalars, $V_\text{eff}(r)$ and $\rho(r)$, such situation is easy for machine learning to learn the mapping from data.

Answer 1

Roughly speaking and provided that the Hohenberg-Kohn theorem applies, if the $v$-representable ground state density of an interacting system $\rho$ is additionally non-interacting $v$-representable, then by definition there exists a non-interacting system with potential $V$ and $\rho$ as its ground state density. By the virtue of the Hohenberg-Kohn theorem, $V$ uniquely determines $\rho$ and vice versa. In the Kohn-Sham approach, then an explicit form of $V=V_\mathrm{eff}=V_\mathrm{eff}[\rho]$ is constructed, such that solving the Kohn-Sham equations self-consistently yields $\rho$.

I don't know what exactly you mean with 'point-wise'; the mapping between $V_\mathrm{eff}$ and $\rho$ is similar to the mapping between the external potential in the interacting system $V_\mathrm{int}$ and $\rho$, i.e. is a mapping between the 'complete' functions, since it is also governed by the Hohenberg-Kohn theorem.