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

S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1980)

$$K= \rho\,e ,$$ (234)

where

$$x=\left (\frac{3}{4\pi\rho}\right )^{1/6} ,$$ (235)

$$\zeta={\frac {\rho_{\alpha}-\rho_{\beta}}{\rho}} ,$$ (236)

$$e=\Lambda+\alpha\,y\left (1+h{\zeta}^{4}\right ) ,$$ (237)

$$y={\frac {9}{8}}\,\left (1+\zeta\right )^{4/3}+{\frac {9}{8}}\,\left (1 -\zeta\right )^{4/3}-\frac{9}{4} ,$$ (238)

$$h={\frac {4\left (\lambda-\Lambda\right )}{9\left ({2}^{1/3}-1\right )\alpha}}-1 ,$$ (239)

$$\Lambda=q(k_{{1}},l_{{1}},m_{{1}},n_{{1}}) ,$$ (240)

$$\lambda=q(k_{{2}},l_{{2}},m_{{2}},n_{{2}}) ,$$ (241)

$$\alpha=q(k_{{3}},l_{{3}},m_{{3}},n_{{3}}) ,$$ (242)

$$q(A,p,c,d)= A\left (\ln \left({\frac {{x}^{2}}{X(x,c,d)}}\right ) +\frac{2\,c}{Q(c,d)}\text{arctan}\left ({ \frac {Q(c,d)}{2\,x+c}}\right )\right.$$

$$\left.-\frac{cp}{X(p,c,d)}\left (\ln \left ({ \frac {\left (x-p\righ... ...t (c+2\,p\right )}{Q(c,d)} \arctan({\frac {Q(c,d)}{2\,x+c}})\right ) \right ) ,$$ (243)

$$Q(c,d)=\sqrt {4\,d-{c}^{2}} ,$$ (244)

$$X(i,c,d)={i}^{2}+ci+d ,$$ (245)

$$k = [ 0.0310907, 0.01554535,-1/\left (6{\pi }\right )^{2}] ,$$ (246)

$$l = [- 0.10498,- 0.325,- 0.0047584] ,$$ (247)

$$m = [ 3.72744, 7.06042, 1.13107]$$ (248)

and

$$n = [ 12.9352, 18.0578, 13.0045] .$$ (249)