17.1.3 VWN: Vosko-Wilk-Nusair (1980) local correlation energy

Can. J. Phys. 58 (1980) 1200. Define:

$$x = \left(3\over4\pi\rho\right)^{1/6}\;; \qquad \zeta = \hat \rho / \rho$$ (10)

Then

$$K(\rho) = \rho \epsilon(x,\zeta)$$ (11)

$$\epsilon(x,\zeta) = \epsilon_P(x) + \alpha(x) g(\zeta) \left(1+h(x)\zeta^4\right)$$ (12)

where

$$g(\zeta) = \frac98 \left( (1+\zeta)^{4/3} + (1-\zeta)^{4/3} -2 \right)$$ (13)

$$h(x) = {4\over 9\left(2^{1/3}-1\right)} {\epsilon_F(x)-\epsilon_P(x)\over\alpha(x)} - 1$$ (14)

and each of $\epsilon_P, \epsilon_F, \alpha$ is a function of the form

$$\epsilon(x) = A\left\{ \ln {x^2\over X(x)} +{2b\over Q} \tan^{-1} {Q\over 2x+b} \right.$$

$$\left. - {bx_0\over X(x_0)} \left[ \ln{\left(x-x_0\right)^2\over X(x)} + {2\left(b+2x_0\right)\over Q} \tan^{-1} {Q\over 2x+b}\right]\right\}$$ (15)

$$X(x) = x^2+bx+c\;;\qquad Q=\left(4c-b^2\right)^{1/2}\;.$$ (16)

with the parameters taking the values
$A=0.0310907, x_0=-0.10498, b=3.72744, c=12.9352$ ($\epsilon_P$);
$A=0.01553535, x_0=-0.32500, b=7.06042, c=18.0578$ ($\epsilon_F$);
$A=-1/6\pi^2, x_0=-0.00475840, b=1.13107, c=13.0045$ ($\alpha$).