Total energy of N+1 electron system compared to N

  1. According to the schematic (FIG 1) in this paper ( the total energy of a solid with N+1 electrons is less (more negative) than the same solid (the same positions of ions) with N electrons. This was explicitly shown for the single fluorine atom (

    My question is that something which can be proven analytically? When I try to simulate this using DFT on a couple of semiconductors , I get the opposite, that this the N system has less energy (more negative) compared to the N+1 system. Any hints are appreciated.

    EDIT: The two articles I cited above are available on arxiv as well, and these are liks:

    Last edited: Aug 5, 2014
  2. jcsd
  3. DrDu

    DrDu 4,639
    Science Advisor

    Which kind of DFT? It is well known that e.g. LDA does not give correct results for the energy gap (i.e. the difference between an N and an N+1 electron system). This is due to the incorrect approximation of the exchange energy. DFT with exact exchange gives correct energy gaps.

    Edit: That is also stated in the second article you cited.
  4. I'm using GGA, but according to the two articles above, the usage of LDA or GGA should not change the sign of E(N)-E(N+1), instead it has an impact on the sign of the 2nd derivative of E(N) when N is regarded as a continuous variable.
  5. Also I should mention that the emphasis here is on localized states. That is the extra one electron in the N+1 system is filling a localized empty electronic state in a solid or an empty orbital in an atom as the in the Fluorine example of the second article of my original post.

    Also when I did the test using GGA (or even GGA+U), I made sure that I have localized states in the semiconductors I examined.
  6. The property in question is the electron affinity, which is indeed positive in most materials (indicating that the energy of the N+1 electron system is less than that of the N electron system). For example, all atoms (except the noble gases) exhibit a positive electron affinity.

    Regarding your DFT calculations, my guess is that the self-interaction error of DFT is what incorrectly raises the energy of the N+1 electron system. If possible, try repeating them with HF (which can be carried out in DFT codes as a limit of hybrid functionals) and see if you at least get the correct sign. For one discussion of this in terms of atoms, see this paper from Kieron Burke's group:
  7. Thank you for your reply and sharing Burke's article, t!m. Indeed for atoms I do get that the energy of the N+1 system is more negative than that of the N system. I got this result on the Fluorine atom using GGA which is in accordance to the second article I posted above.

    However, for a solid that has a lattice distortion (in order to trap a polaron) , I'm not sure why the N+1 system has to have lower energy. I feel that the schematic shown in the first article I posted is for illustration purpose only and it is possible for the N+1 system to have higher energy. But I still need to search more to confirm this.
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