manual   quickstart   instguide   update   basis

Next: C..16 ECERFPBE: Range-Separated Correlation Up: C. Density functional descriptions Previous: C..14 DIRAC: Slater-Dirac Exchange   Contents   Index   PDF


C..15 ECERF: Short-range LDA correlation functional

Local-density approximation of correlation energy
for short-range interelectronic interaction ${\rm erf}(\mu r_{21})/r_{12}$,
S. Paziani, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Phys. Rev. B 73, 155111 (2006).


\begin{displaymath}\nonumber
\epsilon_c^{\rm SR}(r_s,\zeta,\mu) =\epsilon_c^{\rm...
...u^3+a_2 \mu^4+
a_3\mu^5+a_4\mu^6+a_5\mu^8}{(1+b_0^2\mu^2)^4},
\end{displaymath}  

where
\begin{displaymath}
Q(x)=\frac{2\ln(2)-2}{\pi^2}\ln\left(\frac{1+a\,x+b\,x^2+c\,x^3}{1+a\,x+
d\,x^2}\right),
\end{displaymath} (75)

with $a=5.84605$, $c=3.91744$, $d=3.44851$, and $b=d-3\pi\alpha/(4\ln(2)-4)$. The parameters $a_i(r_s,\zeta)$ are given by
$\displaystyle a_1$ $\textstyle =$ $\displaystyle 4 \,b_0^6 \,C_3+b_0^8 \,C_5,$  
$\displaystyle a_2$ $\textstyle =$ $\displaystyle 4 \,b_0^6 \,C_2+b_0^8\, C_4+6\, b_0^4 \epsilon_c^{\rm PW92},$  
$\displaystyle a_3$ $\textstyle =$ $\displaystyle b_0^8 \,C_3,$  
$\displaystyle a_4$ $\textstyle =$ $\displaystyle b_0^8 \,C_2+4 \,b_0^6\, \epsilon_c^{\rm PW92} ,$  
$\displaystyle a_5$ $\textstyle =$ $\displaystyle b_0^8\,\epsilon_c^{\rm PW92},$  

with
$\displaystyle C_2$ $\textstyle =$ $\displaystyle -\frac{3(1\!-\!\zeta^2)\,g_c(0,r_s,\zeta\!=\!0)}{8\,r_s^3}$  
$\displaystyle C_3$ $\textstyle =$ $\displaystyle - (1\!-\!\zeta^2)\frac{g(0,r_s,\zeta\!=\!0)}{\sqrt{2\pi}\, r_s^3}$  
$\displaystyle C_4$ $\textstyle =$ $\displaystyle -\frac{9\, c_4(r_s,\zeta)}{64 r_s^3}$  
$\displaystyle C_5$ $\textstyle =$ $\displaystyle -\frac{9\, c_5(r_s,\zeta)}{40\sqrt{2 \pi} r_s^3}$  
$\displaystyle c_4(r_s,\zeta)$ $\textstyle =$ $\displaystyle \left(\frac{1\!+\!\zeta}{2}\right)^2g''\left(0,
r_s\left(\frac{2}...
...\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ \left(\frac{1\!-\!\zeta}{2}\right)^2 \times$  
    $\displaystyle g''\left(0,
r_s\left(\frac{2}{1\!-\!\zeta}\right)^{1/3}\!\!\!\!\!...
...!=\!1\right)
+ (1\!-\!\zeta^2)D_2(r_s)-\frac{\phi_8(\zeta)}{5\,\alpha^2\,r_s^2}$  
$\displaystyle c_5(r_s,\zeta)$ $\textstyle =$ $\displaystyle \left(\frac{1\!+\!\zeta}{2}\right)^2g''\left(0,
r_s\left(\frac{2}...
...\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ \left(\frac{1\!-\!\zeta}{2}\right)^2 \times$  
    $\displaystyle g''\left(0,
r_s\left(\frac{2}{1\!-\!\zeta}\right)^{1/3}\!\!\!\!\!\!\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ (1\!-\!\zeta^2)D_3(r_s),$ (76)

and
$\displaystyle \phantom{\bigl[} b_0(r_s) = 0.784949\,r_s$     (77)
$\displaystyle \phantom{\Biggl[} g''(0,r_s,\zeta\!=\!1) = \frac{2^{5/3}}{5\,\alpha^2 \,r_s^2} \,
\frac{1-0.02267 r_s}{\left(1+0.4319 r_s+0.04 r_s^2\right)}$     (78)
$\displaystyle \phantom{\Biggl[}D_2(r_s) = \frac{e^{- 0.547 r_s}}{r_s^2}\left(-0.388 r_s+0.676 r_s^2\right)$     (79)
$\displaystyle \phantom{\Biggl[}D_3(r_s) = \frac{e^{-0.31 r_s}}{r_s^3}\left(-4.95 r_s+ r_s^2\right).$     (80)

Finally, $\epsilon_c^{\rm PW92}(r_s,\zeta)$ is the Perdew-Wang parametrization of the correlation energy of the standard uniform electron gas [J.P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992)], and
\begin{displaymath}
g(0,r_s,\zeta\!=\!0)=\frac{1}{2}(1-Br_s+Cr_s^2+Dr_s^3+Er_s^4)\,{\rm e}^{-dr_s},
\end{displaymath} (81)

is the on-top pair-distribution function of the standard jellium model [P. Gori-Giorgi and J.P. Perdew, Phys. Rev. B 64, 155102 (2001)], where $B=-0.0207$, $C=0.08193$, $D=-0.01277$, $E=0.001859$, $d=0.7524$. The correlation part of the on-top pair-distribution function is $g_c(0,r_s,\zeta\!=\!0)=g(0,r_s,\zeta\!=\!0)-\frac{1}{2}$.

Next: C..16 ECERFPBE: Range-Separated Correlation Up: C. Density functional descriptions Previous: C..14 DIRAC: Slater-Dirac Exchange   Contents   Index   PDF

manual   quickstart   instguide   update   basis

molpro@molpro.net 2018-06-18