(This question is originally posted on physics stackexchange, but someone suggested me to post on this site, so there you go)
I'm compiling the mathematical expression of SCAN (Strongly Constrained and Appropriately Normed) functionals' constraints, but apparently they are not very obvious from their paper (at least for me). I have compiled some constraints from the SCAN paper, the PBE paper, and Perdew's presentation, but some are missing (see the last line of this question).
General form
$$ \begin{align} E_{xc}[n] &= \int n \varepsilon_x^{unif}(n) F_{xc}(s,\alpha)\ \mathrm{d}\mathbf{r} \\ E_x[n] &= \int n \varepsilon_x^{unif}(n) F_x(s,\alpha)\ \mathrm{d}\mathbf{r} \\ E_c[n] &= \int n \varepsilon_x^{unif}(n) F_c(r_s,t,\zeta,\alpha)\ \mathrm{d}\mathbf{r} = \int n\left[\varepsilon_c^{unif} + H(r_s,t,\zeta,\alpha)\right]\ \mathrm{d}\mathbf{r} \\ \end{align} $$ where $\varepsilon_x^{unif}(n) = -(3/4\pi)(3\pi^2n)^{1/3}$ and $\varepsilon_c^{unif}$ are obtained from Perdew & Wang, 1992 and the variables $s,\alpha, r_s,t,\zeta$ are listed in SCAN's paper supplementary material.
Exchange constraints
- Negativity $$ F_x(s,\alpha) > 0 $$
- Spin-scaling $$ E_x[n_{\uparrow}, n_{\downarrow}] = \frac{1}{2}\left(E_x[2n_{\uparrow}] + E_x[2n_{\downarrow}]\right) $$
- Uniform density scaling $$ E_x[n_\gamma] = \gamma E_x[n]\mathrm{, where}\ n_\gamma(\mathbf{r}) = \gamma^3 n(\gamma \mathbf{r}) $$
- Fourth order gradient expansion (the expression is from Perdew's presentation) $$ \lim_{s\rightarrow 0, \alpha\rightarrow 1} F_x(s,\alpha) = 1 + \frac{10}{81}s^2 - \frac{1606}{18225} s^4 + \frac{511}{13500} s^2(1-\alpha) + \frac{5913}{405000}(1-\alpha)^2 $$
- Non-uniform density scaling $$ \lim_{s\rightarrow\infty}F_x(s,\alpha) \propto s^{-1/2} $$
- Tight lower bound for two electron densities $$ F_x(s,\alpha=0) \leq 1.174 $$
Correlation constraints
- Non-positivity $$ F_c(r_s,t,\zeta,\alpha) \geq 0 $$
- Second order gradient expansion $$ \begin{align} \lim_{t\rightarrow 0}H(r_s, \zeta, t, \alpha) &= \beta \phi^3 t^2 \\ \beta &\approx 0.066725 \end{align} $$
- Rapidly varying limit (using the term from PBE's paper, instead of from SCAN's paper, is it "Uniform density scaling to the low density limit"?) $$ \lim_{t\rightarrow\infty}H(r_s, \zeta, t, \alpha) = -\varepsilon_c^{unif} $$
- Uniform density scaling to the high density limit $$ \begin{align} \lim_{r_s\rightarrow 0}H(r_s, \zeta, t, \alpha) &= \gamma \phi^3\ln \left(t^2\right) \\ \gamma &= \frac{1}{\pi} (1 - \ln 2) \end{align} $$
- Zero correlation energy for one electron densities $$ H(r_s, \zeta=1, t, \alpha=0) = -\varepsilon_c^{unif} $$
- Finite non-uniform scaling limit (I don't know this)
Exchange and correlation constraints
Size extensivity (I don't know this)
General Lieb-Oxford bound $$ F_{xc}(r_s, \zeta, t, \alpha) \leq 2.215 $$
Weak dependence upon relative spin polarization in the low-density limit (I don't know this)
Static linear response of the uniform electron gas (I don't know this)
Lieb-Oxford bound for two-electron densities $$ F_{xc}(r_s, \zeta=0, t, \alpha=0) \leq 1.67 $$
Summary: What are the constraints for 12, 13, 15, 16? If you want, you can give one constraint in one answer.