C..43 VWN3: Vosko-Wilk-Nusair (1980) III local correlation energy

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C..43 `VWN3`: Vosko-Wilk-Nusair (1980) III local correlation energy

VWN 1980(III) functional See reference [27] for more details.

$\begin{dmath} x=1/4 \sqrt [6]{3}{4}^{5/6}\sqrt [6]{{\frac {1}{\pi \rho}}} ,\end{dmath}$

$\begin{dmath} \zeta={\frac {\rho \left( a \right) -\rho \left( b \right) }{\rho}} ,\end{dmath}$

$\begin{dmath} f=\rho e ,\end{dmath}$

$\begin{dmath} k=[ 0.0310907, 0.01554535,-1/6 {\pi }^{-2}] ,\end{dmath}$

$\begin{dmath} l=[- 0.409286,- 0.743294,- 0.228344] ,\end{dmath}$

$\begin{dmath} m=[ 13.0720, 20.1231, 1.06835] ,\end{dmath}$

$\begin{dmath} n=[ 42.7198, 101.578, 11.4813] ,\end{dmath}$

$\begin{dmath} e=\Lambda+z \left( \lambda-\Lambda \right) ,\end{dmath}$

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

$\begin{dmath} \Lambda=q \left( k_{{1}},l_{{1}},m_{{1}},n_{{1}} \right) ,\end{dmath}$

$\begin{dmath} \lambda=q \left( k_{{2}},l_{{2}},m_{{2}},n_{{2}} \right) ,\end{dmath}$

$\begin{dmath} q \left( A,p,c,d \right) =A \left( \ln \left( {\frac {{x}^{2}}{X ... ...^{-1} \right) \left( X \left( p,c,d \right) \right) ^{-1} \right) ,\end{dmath}$

$\begin{dmath} Q \left( c,d \right) =\sqrt {4 d-{c}^{2}} ,\end{dmath}$

$\begin{dmath} X \left( i,c,d \right) ={i}^{2}+ci+d ,\end{dmath}$

$\begin{dmath} z=4 {\frac {y}{9 \sqrt [3]{2}-9}} .\end{dmath}$

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