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Total energy of N+1 electron system compared to N

  1. Aug 5, 2014 #1
    According to the schematic (FIG 1) in this paper (http://journals.aps.org/prb/abstract/10.1103/PhysRevB.80.085202) 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 (http://journals.aps.org/pra/abstract/10.1103/PhysRevA.76.040501).

    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:

    (1) http://arxiv.org/pdf/0905.0018.pdf
    (2) http://arxiv.org/pdf/cond-mat/0702283v2.pdf
    Last edited: Aug 5, 2014
  2. jcsd
  3. Aug 5, 2014 #2


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    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. Aug 5, 2014 #3
    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. Aug 6, 2014 #4
    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. Aug 10, 2014 #5


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    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: http://dft.uci.edu/pubs/LFB10.pdf
  7. Aug 12, 2014 #6
    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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