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In my experience with (Kohn-Sham) density functional theory, the external potential is due to a massive particle like a nucleus or ion. I have never seen an electron be the source of the potential, and I wonder why. For instance, if I want to study the electron gas model using DFT, it seems entirely natural to imagine the gas's response to an additional electron placed at the origin. However, I can imagine two objections to this procedure:

  • The Heisenberg uncertainty principle prevents us from knowing that the inserted electron is both at the origin and at rest.
  • Because electrons are identical fermions, we are not allowed to single one of them out as being the source of the external potential.

Neither of these are practical issues in the case where the external potential is due to a heavy particle. Do either of these objections (or possibly others that I haven't thought of) mean that an external potential in DFT cannot be sourced by an electron?

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In DFT, the electron-electron interactions are contained in the so-called Exchange-Correlation functional which is a functional of the electronic density and it's derivatives. This functional is a static approximation (i.e. time averaged) of the potential an electron feels from the other electrons. It is thus a time-averaged mean field approximation.

The term "external potential" is thus reserved for the ionic potentials from the atomic cores in a system. It would thus be set to 0 in the particular case of an electron gas.

In usual implementations of DFT, we consider a fixed number of particles. If one wants to add an electron to a system (like for doping), one either builds supercells (in case of crystals) or one adds a static charge background. In the particular case of adding one electron to a single molecule/gas, my reflex would be to compute both the case without the added electron and the case with it and then see the effects of the added electron on the quantities of interest.

Finally, for the Heisenberg uncertainty principle, in atomic calculations, one usually considers orbitals or wave packets. In those cases, the particle's position is not clearly defined but it's energy is well known and that is the only relevant quantity in DFT since we need the particle's energy to determine the total energy of the system in order to minimize it.

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  • $\begingroup$ The answer makes enough sense, but I think it does not address the question, which is one of basic physics not "how to calculate the effect of +1 electron". I want to know if there is some physical principle that forbids using a stationary electron as a test particle. $\endgroup$ – Endulum Jul 5 at 15:22
  • $\begingroup$ Well the answer is no as you can indeed compute the effect of adding an electron to a system as stated in the answer. But in Quantum Mechanics there is no such thing as a stationary point particle, only particles and their wave function. $\endgroup$ – fgoudra Jul 5 at 16:12

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