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Next: C..44 PBESOLC: PBEsol Correlation Up: C. Density functional descriptions Previous: C..42 P86: .   Contents   Index   PDF


C..43 PBEC: PBE Correlation Functional

\begin{dmath}
f=\rho\, \left( \epsilon \left( \rho \left( a \right) ,\rho \left(...
...ft( d,\rho \left( a \right) ,\rho \left( b
\right) \right) \right)
,\end{dmath}

\begin{dmath}
G=\rho\, \left( \epsilon \left( \rho \left( s \right) ,0 \right) +C
\left( Q,\rho \left( s \right) ,0 \right) \right)
,\end{dmath}

\begin{dmath}
d=1/12\,{\frac {\sqrt {\sigma}{3}^{5/6}}{u \left( \rho \left( a
\...
...\rho \left( b \right) \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) =1/2\, \left( u \left( \rho \left(...
...,\beta \right) \right) ^{2}{d}
^{4} \right) }} \right) {\iota}^{-1}
,\end{dmath}

\begin{dmath}
A \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-...
...t( b \right) \right) \right) ^{3}{
\lambda}^{2}}}}}-1 \right) ^{-1}
,\end{dmath}

\begin{dmath}
\iota= 0.0716
,\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}
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}

\begin{dmath}
P=[1,1,1]
,\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
\...
...1}{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 \left( {\it ss} \right) }\sqrt [3]{2}{3}^
{5/6}}{\sqrt [6]{{\pi }^{-1}}{\rho}^{7/6}}}
.\end{dmath}



molpro@molpro.net 2018-01-18