# Behavior of the Vosco-Wilk-Nusair correlation functional

Assumptions:

1. interacting system

2. uniform gas system

In the Kohn-Sham model, the exchange correlation functional is introduced to account for correlation energy $$\epsilon_{xc} = \epsilon_{x} + \epsilon_{c}$$ is the exchange correlation energy per particle for a uniform electron gas where $$\epsilon_{x}$$ is the exchange energy per particle for a uniform electron gas and $$\epsilon_{c}$$ is the correlation energy per particle for a uniform electron gas.

The correlation energy of an interacting electron gas system has analytic solutions only in the limiting cases:

$$\lim_{\rho \rightarrow 0}: \epsilon_{c} (\rho) = 0.311ln(r_{s}) - 0.048 + r_{s}(A^{0}ln(r_{s})++C^{0}) \rightarrow r_{s} >> 1$$

$$\lim_{\rho \rightarrow \infty}: \epsilon_{c}(\rho)=0.5[\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + ...]$$

However, for regions within the bounds of the limiting case, the Vosco-Wilk-Nusair correlation functional is invoked.

$$\epsilon_{c}(r_{s})=\frac{A}{2}[ln(\frac{x}{X(x)} + \frac{2b}{Q}tan^{-1}(\frac{Q}{2x+b})-\frac{bx_{0}}{X(x_{0})}[ln(\frac{(x-x_{0})^{2}}{X(x)})+\frac{2(b+2x_{0})}{Q}tan^{-1}(\frac{Q}{2x+b})]]$$

What does $$A, r_{s}, x, X(x), Q$$ refers to?

A very quick google shows up https://math.nist.gov/DFTdata/atomdata/node5.html#SECTION00021200000000000000 amongst other hits - note the definition is slightly different from yours and I think you want $$x^2$$ in the first term above, see also https://www.molpro.net/manual/doku.php?id=density_functional_descriptions. Quoting the first, with images where appropriate as it will be non-trivial to reformat:
where we have $$x={r_s}^{1 \over 2}$$, $$X( x )=x^2+bx+c$$, $$Q=(4c-b^2)^{1 \over 2}$$. The parameters $$x_0$$, b and c, given in the table below, are used to create three instances of F, using the table below.
$$r_s$$ is described on https://math.nist.gov/DFTdata/atomdata/node4.html#SECTION00021100000000000000. Again quoting with images hwere appropriate
The electron gas parameter $$r_s$$ ... are defined as