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C..30 PW91C: Perdew-Wang 1991 GGA Correlation Functional

See reference [5] for more details.

\begin{dmath} K= \rho\, \left( \epsilon \left( \rho_{\alpha},\rho_{\beta} \right) +H \left( d,\rho_{\alpha},\rho_{\beta} \right) \right) ,\end{dmath} where \begin{dmath} d=1/12\,{\frac {\sqrt {\sigma}{3}^{5/6}}{u \left( \rho_{\alpha}, \rho_{\beta} \right) \sqrt [6]{{\pi }^{-1}}{\rho}^{7/6}}} ,\end{dmath} \begin{dmath} u \left( \alpha,\beta \right) =1/2\, \left( 1+\zeta \left( \alpha,... ...3}+1/2\, \left( 1-\zeta \left( \alpha,\beta \right) \right) ^{2/3} ,\end{dmath} \begin{dmath} H \left( d,\alpha,\beta \right) =L \left( d,\alpha,\beta \right) +J \left( d,\alpha,\beta \right) ,\end{dmath} \begin{dmath} L \left( d,\alpha,\beta \right) =1/2\, \left( u \left( \rho_{\alph... ...,\beta \right) \right) ^{2}{d}^{4} \right) }} \right) { \iota}^{-1} ,\end{dmath} \begin{dmath} J \left( d,\alpha,\beta \right) =\nu\, \left( \phi \left( r \left(... ...ight) \right) ^{4}{3}^{2/3}{d}^{2}}{\sqrt [3]{{\pi }^{5}\rho}}}}} ,\end{dmath} \begin{dmath} A \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-... ...\rho_{\beta} \right) \right) ^{3}{\lambda}^{2}}}}}-1 \right) ^{-1} ,\end{dmath} \begin{dmath} \iota= 0.09 ,\end{dmath} \begin{dmath} \lambda=\nu\,\kappa ,\end{dmath} \begin{dmath} \nu=16\,{\frac {\sqrt [3]{3}\sqrt [3]{{\pi }^{2}}}{\pi }} ,\end{dmath} \begin{dmath} \kappa= 0.004235 ,\end{dmath} \begin{dmath} Z=- 0.001667 ,\end{dmath} \begin{dmath} \phi \left( r \right) =\theta \left( r \right) -Z ,\end{dmath} \begin{dmath} \theta \left( r \right) ={\frac {1}{1000}}\,{\frac { 2.568+\Xi\,r+\Phi \,{r}^{2}}{1+\Lambda\,r+\Upsilon\,{r}^{2}+10\,\Phi\,{r}^{3}}} ,\end{dmath} \begin{dmath} \Xi= 23.266 ,\end{dmath} \begin{dmath} \Phi= 0.007389 ,\end{dmath} \begin{dmath} \Lambda= 8.723 ,\end{dmath} \begin{dmath} \Upsilon= 0.472 ,\end{dmath} \begin{dmath} \epsilon \left( \alpha,\beta \right) =e \left( r \left( \alpha,\be... ...ght) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{4} ,\end{dmath} \begin{dmath} r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,\end{dmath} \begin{dmath} \zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,\end{dmath} \begin{dmath} \omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,\end{dmath} \begin{dmath} e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln ... ...}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,\end{dmath} \begin{dmath} c= 1.709921 ,\end{dmath} \begin{dmath} C \left( d,\alpha,\beta \right) =K \left( Q,\alpha,\beta \right) +M \left( Q,\alpha,\beta \right) ,\end{dmath} \begin{dmath} M \left( d,\alpha,\beta \right) = 0.5\,\nu\, \left( \phi \left( r ... ...35.9789467\,{\frac {{3}^{2/3}{d}^{2}}{\sqrt [3]{{\pi }^{5}\rho}}}}} ,\end{dmath} \begin{dmath} K \left( d,\alpha,\beta \right) = 0.2500000000\,{\lambda}^{2}\ln ... ...,\beta \right) \right) ^{2}{d} ^{4} \right) }} \right) {\iota}^{-1} ,\end{dmath} \begin{dmath} N \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-... ...on \left( \alpha,\beta \right) }{{\lambda}^{2}}}}}- 1 \right) ^{-1} ,\end{dmath} \begin{dmath} Q=1/12\,{\frac {\sqrt {\sigma_{ss}}\sqrt [3]{2}{3}^{5/6}}{\sqrt [6]{{ \pi }^{-1}}{\rho}^{7/6}}} ,\end{dmath} \begin{dmath} T = [ 0.031091, 0.015545, 0.016887] ,\end{dmath} \begin{dmath} U = [ 0.21370, 0.20548, 0.11125] ,\end{dmath} \begin{dmath} V = [ 7.5957, 14.1189, 10.357] ,\end{dmath} \begin{dmath} W = [ 3.5876, 6.1977, 3.6231] ,\end{dmath} \begin{dmath} X = [ 1.6382, 3.3662, 0.88026] ,\end{dmath} \begin{dmath} Y = [ 0.49294, 0.62517, 0.49671] \end{dmath} and \begin{dmath} P = [1,1,1] .\end{dmath} To avoid singularities in the limit $\rho_{\bar{s}}\rightarrow 0$ \begin{dmath} G= \rho\, \left( \epsilon \left( \rho_{s},0 \right) +C \left( Q,\rho_{s},0 \right) \right) .\end{dmath}

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Oct 10, 2007