19.1.21 P86:

J. P. Perdew, Phys. Rev. B 33, 8822 (1986)

VWN with gradient correction.

$$K= \rho\,e+{\frac {{e^{-\Phi}}C(r)\sigma}{d{\rho}^{4/3}}} ,$$ (95)

where

$$x=\left ({{\frac {3}{4 \pi \rho}}}\right )^{1/6} ,$$ (96)

$$\zeta={\frac {\rho_{\alpha}-\rho_{\beta}}{\rho}} ,$$ (97)

$$e=\Lambda+\omega\,y\left (1+h{\zeta}^{4}\right ) ,$$ (98)

$$y={\frac {9}{8}}\,\left (1+\zeta\right )^{4/3}+{\frac {9}{8}}\,\left (1 -\zeta\right )^{4/3}-\frac{9}{4} ,$$ (99)

$$h={\frac {4\left (\lambda-\Lambda\right )}{9\left ({2}^{1/3}-1\right )\omega}}-1 ,$$ (100)

$$\Lambda=q(k_{{1}},l_{{1}},m_{{1}},n_{{1}}) ,$$ (101)

$$\lambda=q(k_{{2}},l_{{2}},m_{{2}},n_{{2}}) ,$$ (102)

$$\omega=q(k_{{3}},l_{{3}},m_{{3}},n_{{3}}) ,$$ (103)

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

$$\left.-\frac{cp}{X(p,c,d)}\left (\ln \left ({ \frac {\left (x-p\righ... ...t (c+2\,p\right )}{Q(c,d)} \arctan({\frac {Q(c,d)}{2\,x+c}})\right ) \right ) ,$$ (104)

$$Q(c,d)=\sqrt {4\,d-{c}^{2}} ,$$ (105)

$$X(i,c,d)={i}^{2}+ci+d ,$$ (106)

$$\Phi= 0.007390075\,{\frac {z\sqrt {\sigma}}{C(r){\rho}^{7/6}}} ,$$ (107)

$$d={2}^{1/3}\sqrt {\left (1/2+\zeta/2\right )^{5/3}+\left (1/2-\zeta/2\right )^{5/3}} ,$$ (108)

$$C(r)= 0.001667+{\frac { 0.002568+\alpha\,r+\beta\,{r}^{2}}{1+\xi\,r+ \delta\,{r}^{2}+10000\,\beta\,{r}^{3}}} ,$$ (109)

$$z= 0.11 ,$$ (110)

$$\alpha= 0.023266 ,$$ (111)

$$\beta= 0.000007389 ,$$ (112)

$$\xi= 8.723 ,$$ (113)

$$\delta= 0.472 ,$$ (114)

$$k = [ 0.0310907, 0.01554535,-1/\left (6{\pi }\right )^{2}] ,$$ (115)

$$l = [- 0.10498,- 0.325,- 0.0047584] ,$$ (116)

$$m = [ 3.72744, 7.06042, 1.13107]$$ (117)

and

$$n = [ 12.9352, 18.0578, 13.0045] .$$ (118)