Related question: Mathematical expression of SCAN (Strongly Constrained and Appropriately Normed) constraints in DFT

The SCAN (strongly-constrained and appropriately-normed) meta-GGA constructed in 2015 is the most promising metaGGA which respects all 17 known exact constraints that a meta-GGA can satisfy. What are these constraints? What's the significance?


2 Answers 2


I list these 17 constraints directly from 2015 Sun et. al. paper:

For exchange potential:

  1. Negativity.
  2. Spin-scaling.
  3. Uniform density scaling.
  4. Fourth-order gradient expansion.
  5. Non-uniform density scaling.
  6. Tight bounds for two-electron densities.

For correlation potential:

  1. Non-positivity.
  2. Second-order gradient expansion.
  3. Uniform density scaling to the high-density limit.
  4. Uniform density scaling to the low-density limit.
  5. Zero correlation energy for any one-electron spin-polarized density.
  6. Non-uniform density scaling.

For both exchange and correlation potentials:

  1. Size extensively.
  2. General Lieb-Oxford bound.
  3. Weak dependence upon relative spin polarization in the low-density limit.
  4. Static linear response of the uniform electron gas.
  5. Lieb-Oxford bound for two-electron densities.

Some of these conditions are based on properties of exact $E_{\text{XC}}[n(\mathbf{r})]$ for example we know that correlation potential should be self-correlation free for one-electron (constraint 11), etc. Keep in mind that there are infinitely many ways to satisfy these conditions and it's not guaranteed if an exchange-correlation model satisfies these constraints, it's suitable for all applications.


This is more of supplement to Alone Programmer's answer to expand a bit more on the significance of meeting these constraints.

These constraints allow the SCAN functional to reproduce the results of the true density functional for simplified model system. This agreement suggests that the form of the functional is more physically consistent than a functional that does not follow these constraints. In principle, agreeing with the true density functional should make SCAN more robust than other functionals and better able to describe systems that it wasn't specifically parameterized for.


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