Next: 17.1.4 LYP: Lee-Yang-Parr (1988) Up: 17.1 Density Functionals Previous: 17.1.2 B: Becke (1988)


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

Can. J. Phys. 58 (1980) 1200. Define:
$\displaystyle x$ $\textstyle =$ $\displaystyle \left(3\over4\pi\rho\right)^{1/6}\;;
\qquad
\zeta = \hat \rho / \rho$ (10)

Then
$\displaystyle K(\rho)$ $\textstyle =$ $\displaystyle \rho \epsilon(x,\zeta)$ (11)
$\displaystyle \epsilon(x,\zeta)$ $\textstyle =$ $\displaystyle \epsilon_P(x) + \alpha(x) g(\zeta) \left(1+h(x)\zeta^4\right)$ (12)

where
$\displaystyle g(\zeta)$ $\textstyle =$ $\displaystyle \frac98 \left( (1+\zeta)^{4/3} + (1-\zeta)^{4/3} -2 \right)$ (13)
$\displaystyle h(x)$ $\textstyle =$ $\displaystyle {4\over 9\left(2^{1/3}-1\right)}
{\epsilon_F(x)-\epsilon_P(x)\over\alpha(x)} - 1$ (14)

and each of $\epsilon_P, \epsilon_F, \alpha$ is a function of the form
$\displaystyle \epsilon(x)$ $\textstyle =$ $\displaystyle A\left\{ \ln {x^2\over X(x)}
+{2b\over Q} \tan^{-1} {Q\over 2x+b}
\right.$  
    $\displaystyle \left.
- {bx_0\over X(x_0)}
\left[ \ln{\left(x-x_0\right)^2\over X(x)}
+ {2\left(b+2x_0\right)\over Q}
\tan^{-1} {Q\over 2x+b}\right]\right\}$ (15)
$\displaystyle X(x)$ $\textstyle =$ $\displaystyle x^2+bx+c\;;\qquad Q=\left(4c-b^2\right)^{1/2}\;.$ (16)

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

Next: 17.1.4 LYP: Lee-Yang-Parr (1988) Up: 17.1 Density Functionals Previous: 17.1.2 B: Becke (1988)

P.J. Knowles and H.-J. Werner
molpro-support@tc.bham.ac.uk
Mar 8, 2000