# Mathematical expression of SCAN (Strongly Constrained and Appropriately Normed) constraints in DFT

(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

1. Negativity $$F_x(s,\alpha) > 0$$
2. Spin-scaling $$E_x[n_{\uparrow}, n_{\downarrow}] = \frac{1}{2}\left(E_x[2n_{\uparrow}] + E_x[2n_{\downarrow}]\right)$$
3. Uniform density scaling $$E_x[n_\gamma] = \gamma E_x[n]\mathrm{, where}\ n_\gamma(\mathbf{r}) = \gamma^3 n(\gamma \mathbf{r})$$
4. Fourth order gradient expansion (the expression is from Perdew's presentation) $$\lim_{s\rightarrow 0, \alpha\rightarrow 1} F_x(s,\alpha) = 1 + \frac{10}{81}s^2 - \frac{1606}{18225} s^4 + \frac{511}{13500} s^2(1-\alpha) + \frac{5913}{405000}(1-\alpha)^2$$
5. Non-uniform density scaling $$\lim_{s\rightarrow\infty}F_x(s,\alpha) \propto s^{-1/2}$$
6. Tight lower bound for two electron densities $$F_x(s,\alpha=0) \leq 1.174$$

Correlation constraints

1. Non-positivity $$F_c(r_s,t,\zeta,\alpha) \geq 0$$
2. 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}
3. 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"?) $$\lim_{t\rightarrow\infty}H(r_s, \zeta, t, \alpha) = -\varepsilon_c^{unif}$$
4. 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}
5. Zero correlation energy for one electron densities $$H(r_s, \zeta=1, t, \alpha=0) = -\varepsilon_c^{unif}$$
6. Finite non-uniform scaling limit (I don't know this)

Exchange and correlation constraints

1. Size extensivity (I don't know this)

2. General Lieb-Oxford bound $$F_{xc}(r_s, \zeta, t, \alpha) \leq 2.215$$

3. Weak dependence upon relative spin polarization in the low-density limit (I don't know this)

4. Static linear response of the uniform electron gas (I don't know this)

5. 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.

• +10. Great question! We are very happy to see you here, and hope to see you more !!! Jun 17, 2020 at 16:44
• Thanks! How do you judge if a certain functional satisfy the size extensivity constraint? Is being semi-local (e.g. LDA, GGA, MetaGGA) automatically satisfy the size extensivity? Jun 17, 2020 at 20:12

## Constraint #13: Size-Extensivity

While the Wikipedia page for size-consistency and size-extensivity gives a clear formula for the definition of size-consistency, unfortunately they did not give a definition of size-extensivity, so I had to look deeper into the reference that they provided. They say that size-extensivity was introduced by Bartlett, and they cite this review paper of his from 1981, but this paper itself credits the following papers, which I have now looked at for the first time and summarized below:

• (1955) Keith Brueckner first recognized in his study of the uniform electron gas, that some terms in the energy obtained by Rayleigh-Schroedinger perturbation theory, incorrectly do not scale linearly with the number of electrons $$N$$ as $$N\rightarrow \infty$$. He found a way to cancel out all of these spurious terms, up to fourth-order in the perturbation theory. These spurious terms are also the reason why CI (configuration interaction) converges slowly with respect to the number of excitations included. A year later, Brueckner published the paper with Gell-Mann that became the subject of my other answer here. He was one of the all-time greats, and lived to 90.
• (1957) Jeffrey Golstone proved the "linked-diagram theorem" which ensures that the spurious terms found by Brueckner, get cancelled out to all orders in perturbation theory. Goldstone by the way, is one of the most influential physicists that's still alive! He's currently 85 and he was even a co-author on the quite recent paper that popularized Adiabatic Quantum Computing :)
• (1965) Bartlett's review paper says that Hans Primas was actually the one to first really emphasize this concept of having proper scaling. I don't know much about Primas, though I found that he survived to the age of 86 :)
• (1971-1973) Wilfried Meyer used this concept of size-extensivity to justify the CEPA model. At the time, Meyer had just finished making the MOLPRO software in the 1960s, a software which is now more than 50 years later, maybe the most popular quantum chemistry software for fast high-accuracy calculations.
• (1978) Bartlett and Purvis used the term "size-extensivity" here, so this is perhaps where the term was first introduced, but he uses it to describe what the 1955 and 1957 papers achieved.

So what is size-extensivity?

My reading of the above Bartlett papers tells me that for a homogenous system like an electron gas or a set of non-interacting He atoms, the energy should scale linearly with the number of particles and that the concept can also be generalized to properties other than energy.

• Exactly, twice as much stuff should have twice the energy. It might sound obvious, but not all functionals have this property, e.g. the Tran-Blaha functional (although that is a little unfair as an example, since it wasn't ever intended as a "proper" functional). Jun 29, 2020 at 2:11

## Constraints #9, #10, #12: Uniform and non-uniform density scaling limit

The uniform density scaling constraint is obtained by substituting the density into $$n(\mathbf{r}) \rightarrow n_\gamma (\mathbf{r})$$ which is defined by $$n_\gamma(\mathbf{r}) = \gamma^3 n(\gamma\mathbf{r}).$$ While the non-uniform density scaling is given as \begin{align} n^x_\gamma(x,y,z) &= \gamma n(\gamma x, y, z)\\ n^{xy}_{\gamma\gamma}(x,y,z) &= \gamma^2 n(\gamma x, \gamma y, z) \end{align}

### Constraint #9: Uniform density scaling to the low density limit

[SCAN paper 1st paragraph page 3] "It properly scales to ... zero like the exchange energy in the low-density limit", so I guess what it says is $$\lim_{\gamma\rightarrow 0}E_c[n_\gamma] = 0$$

### Constraint #10: Uniform density scaling to the high density limit

[SCAN paper 1st paragraph page 3] "It properly scales to a finite negative value per electron under uniform density scaling to the high-density limit" which then cites the reference Levy, 1990 containing the mathematical expression (i.e. eq. 12), $$\lim_{\gamma\rightarrow\infty} E_c[n_\gamma] > -\infty$$ The expression in the question is the implication of the equation above to make the correlation energy finite at limit $$\gamma\rightarrow\infty$$. This is because the correlation energy for uniform electron gas scales with $$\ln(r_s)$$ at this limit and can reach $$-\infty$$, therefore $$H$$ should have the factor of $$\ln(t^2)$$ to cancel this effect and keep the correlation energy finite at this limit.

### Constraint #12: Finite non-uniform scaling limit

[SCAN paper 1st paragraph on page 3] "Its correlation energy per electron is properly finite (but improperly zero) under nonuniform density scaling to the true two-dimensional limit", referencing Levy, 1990 which contains in eq. 30 & 31: \begin{align} \lim_{\gamma\rightarrow\infty}\gamma^{-1} E_c[n_{\gamma\gamma}^{xy}] &> -\infty \\ \lim_{\gamma\rightarrow 0}E_c[n_{\gamma}^x] &> -\infty \end{align}

## Constraint #16: Static linear response of the uniform electron gas

This constraint is actually one of the 11 exact constraints that are also satisfied by the PBE GGA functional, as well as the TPSS Meta-GGA. If I'm understanding [1] correctly, the form of this constraint is:

$$\lim_{k\to0}\gamma_x(y)=1+\frac{5}{9}y^2+\frac{73}{225}y^4-\frac{146}{3375}y^6+\mathcal{O}(y^8)$$

where $$\gamma_x(y)$$ is the response function of jellium/uniform electron gas and $$y=\frac{k}{2k_F}$$ with $$k$$ the perturbation wavevector and $$k_F$$ the Fermi energy per electron.

Clearly this is a desirable trait for our any given density functional approximation to have: if we can't even correctly predict how the density of an electrons cloud responds to a vanishingly small perturbation, we will likely have a challenging time predicting the effect on a molecular system or solid where we have to deal with the potential coming from discrete positive charges (i.e. nuclei).

This is not to say that functional that fail to satisfy this constraint can't be accurate for a great many cases. However, failing to meet this constraint (or any of these exact constraints) does suggest that a functional could be less generalizable and less accurately describing the true underlying physics.

### References:

1. Tao, J.; Perdew, J. P.; Almeida, L. M.; Fiolhais, C.; Kümmel, S. Nonempirical density functionals investigated for jellium: Spin-polarized surfaces, spherical clusters, and bulk linear response. Phys. Rev. B 2008, 77 (24), No. 245107. DOI: 10.1103/PhysRevB.77.245107.

## Constraint #15: Weak dependence upon relative spin polarization in the low-density limit

[SCAN paper 2nd paragraph page 3] "... in the low-density limit, where our $$F_{xc}$$ properly shows a weak dependence on relative spin polarization" $$\lim_{r_s\rightarrow \infty} F_{xc}(r_s, \zeta_1, t, \alpha) \approx \lim_{r_s\rightarrow \infty} F_{xc}(r_s, \zeta_2, t, \alpha)$$

• Does this really need to be an answer? It's already in the question. Jun 18, 2020 at 15:24
• Ideally we would not have one person answering multiple times like this: materials.meta.stackexchange.com/a/126/5. It's better to pick one constraint and explain it in thorough detail! Another person can pick another constraint, and explain that one in thorough detail! It will bring in a diversity of experts and give a lot of people an opportunity to participate. Jun 18, 2020 at 15:39
• @NikeDattani It looks like it was added to the question afterwards as an edit. It would probably be better if the question remained fixed and any additional information was given in answers.
– Tyberius
Jun 18, 2020 at 16:15
• I've revert it back to the semi-original question to avoid confusion like this Jun 18, 2020 at 16:27
• I know its been awhile since this question/answer were posted, but I think this answer could be improved with a little more detail. What is the interpretation of this equation? What effect does this constraint have on the calculations done with SCAN? Does it help/hurt accuracy for a particular class of compounds?
– Tyberius
Oct 1, 2021 at 13:44