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I calculated the electron density of a molecule by a DFT software (Gaussian 16) with a standard setting in terms of the basis set and hybrid functional. After that calculation, I viewed the plot about the correspondence between the obtained electron density $\rho$ and the external potential $V$ by Coulomb (i.e., $V(r) = -\sum_{i=1}^{M} \frac{Z_i}{||r-R_i||}$, where $Z$ is the atomic number and $R$ is the atomic position) on some grid points $r$ in the molecule.

The Hohenberg-Kohn theorem shows the one-to-one correspondence between $\rho$ and $V$; however, my calculation result did not show it. The following figure shows that the same density values provided different potential values or the same potential values provided different density values.

My calculation might be wrong, but I have tried to calculate many times, and the result seems to be correct. Why did I get such a result that seems to contradict the Hohenberg-Kohn theorem?

enter image description here

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This situation does not contradict the Hohenberg-Kohn theorems:

the same density values provided different potential values or the same potential values provided different density values.

The first Hohenberg-Kohn theorem refers not to pointwise correspondence, but rather to correspondence between entire functions. It tells us that if I give you the full density—the entire function, not just values at a few points—you can (in principle) determine the shape of the full potential. However, the relationship is complex: the density at a single point will likely depend on the potential at all nearby points, and vice versa.

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  • $\begingroup$ Thank you for the answer. I still do not understand that "not pointwise correspondence, but correspondence between entire functions." I believe that correspondence as entire functions means (encompasses) the pointwise correspondence, but is it not right? $\endgroup$
    – neco
    Commented Jul 29, 2021 at 14:25
  • $\begingroup$ Imagine a secret code where 'apple' means 'water.' The H-K correspondence works similarly: correspondence between functions is like correspondence between words. But the individual letters in 'apple' and 'water' do not have any correspondence; otherwise 'p' would have to correspond to both 'a' and 't'. $\endgroup$
    – wcw
    Commented Jul 29, 2021 at 15:41
  • $\begingroup$ I'm even more confused... In a popular figure about one-to-one correspondence (e.g., en.wikipedia.org/wiki/Bijection), your example seems that all arrows are completely destroyed. Do we still call such situation one-to-one correspondence? $\endgroup$
    – neco
    Commented Jul 29, 2021 at 18:39
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    $\begingroup$ Yes, correspondence between the functions themselves $\endgroup$
    – wcw
    Commented Jul 29, 2021 at 20:56
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    $\begingroup$ If the discussion gets too much longer, I would recommend this chat room. $\endgroup$
    – Nike Dattani - No Free Time
    Commented Jul 29, 2021 at 21:35

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