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This question is inspired by this post. In the Kohn-Sham framework of density functional theory, the total energy is expressed as: $$E=E_{kin}^{non}+E_{ext}+E_{H}+E_{xc}$$ in which

  • The first term is the kinetic energy of the non-interacting system: $$E^{non}_{kin}=-\dfrac{1}{2}\sum_i^{n}\phi_i^*(\vec{r})\nabla^2\phi_i(\vec{r})$$

  • The second term is the external energy: $$E_{ext}=\int\phi^*(\vec{r})U_{ext}(\vec{r})\phi(\vec{r}) d\vec{r}=\int U_{ext}(\vec{r})\rho(\vec{r})d\vec{r}$$

  • The third term is the Hartree energy: $$E_H=\dfrac{1}{2}\int \int \dfrac{\rho(\vec{r})\rho(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}d\vec{r}'$$

  • The last term is the exchange-correlation term needs to be approximated, such as LDA, which includes:

    • Exchange energy;
    • Correlation energy;
    • The kinetic energy difference between the interacting system and the non-interacting system;
    • The self-interaction error in the Hartree energy.

In this post, Phil Hasnip has claimed that including a Hubbard U potential within a Kohn-Sham density functional theory (DFT) calculation is to study a material for which you expect a significant self-interaction error.

So is there any other interesting physics contained in the exchange-correlation functional in the framework of KS-DFT?

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You have already detailed what the physics contained in the exchange-correlation functional is.

Self-interaction error is not physical; it is an artifact in density functional approximations that arises from imperfect cancellation of the Coulomb and exchange interactions. Although several schemes have been suggested to remove self-interaction error, the one by Perdew and Zunger being the most well-known, these schemes tend to be problematic in that they are

  • hideously expensive computationally
  • hard to minimize properly (e.g. the Perdew-Zunger functional requires complex orbitals even for a gas-phase atom without external fields)
  • may break molecular symmetries

Another failure of density functional approximations is in modeling systems with strong correlations, such as many transition metal complexes and multiple bond breaking.

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