The Hartree-Fock equations include a term for the exchange interaction, which is usually explained as a repulsive force due to the Pauli exclusion principle. (It says so right in the description for the "exchange-interaction" tag.)

I expect that, in a system of repulsive particles, increasing the particle density would increase the energy of the system. (E.g. if you have a bunch of particles connected by springs, you would have to work to compress the system, so the energy of the system increases when compressed.)

However, for a free electron gas (and for the local density approximation), the exchange energy becomes more negative as the density is increased [Ashcroft and Mermin eq. (17.26)]:

$$U_{ex} \propto -\left(n\left(\vec{r}\right)\right)^{1/3},$$

where $n\left(\vec{r}\right)$ is the electron density. [See also Ashcroft and Mermin eq. (17.24) and the two sections on Hartree-Fock.] I interpret this to mean that increasing the electron density results in lower energies due to the exchange interaction.

Why does the exchange interaction -- a repulsive force -- lead to lower energy as the particle density is increased?


1 Answer 1


The exchange interaction includes not only a term reflecting the repulsive Pauli interaction, but also a correction for the electron self-energy of the Hartree equations. That is because the electron density that is used to calculate the Hartree potential includes the density of the orbital of the electron under consideration as well as all the other electron orbitals. The Fock exchange term thereby includes a term that removes this self-energy. The self-energy is a repulsion, so the terms that corrects for it is an attractive interaction. If this dominates, then the overal Fock term will be attractive and will increase as the electron density increases.

  • $\begingroup$ Do you have any references on this that you could share? $\endgroup$
    – lnmaurer
    Jan 21, 2018 at 22:10
  • $\begingroup$ You may find a discussion of this issue in John Slater's text "Quantum Theory of Atonic Structure" Vol II on page 8. I remembered this from Slater's lectures (1967, Univ of Florida), but still have the text and looked up the reference. BTW the approximate local exchange interaction that you give above was first derived by Slater. The nonlocal exchange interaction that includes both the Pauli repulsion and the attractive self-energy correction may differ somewhat from the local approximation. Hence, I am not certain that the exchange energy correction is always attractive. $\endgroup$ Jan 22, 2018 at 0:30
  • $\begingroup$ yes it is: chemistry.stackexchange.com/questions/40282/… $\endgroup$
    – cipper
    Mar 30 at 16:43

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