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

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.

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.
• Thank you very much for the quick reply. The word of 'point-wise' mapping means the mapping between the value of $V_{eff}(r)$ on a point $r$ and the value of $\rho(r)$ on $r$, not the the mapping as entire functions, $V_{eff}$ and $\rho$ in the system. I have added the postscript.