# Density functional descriptions

Divergence free semiempirical gradient-corrected exchange energy functional. $\lambda=\gamma$ in ref. $$g=-{\frac {c \left( \rho \left( s \right) \right) ^{4/3} \left( 1+ \beta\, \left( \chi \left( s \right) \right) ^{2} \right) }{1+\lambda \, \left( \chi \left( s \right) \right) ^{2}}} ,$$

$$G=-{\frac {c \left( \rho \left( s \right) \right) ^{4/3} \left( 1+ \beta\, \left( \chi \left( s \right) \right) ^{2} \right) }{1+\lambda \, \left( \chi \left( s \right) \right) ^{2}}} ,$$

$$c=3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$\beta= 0.0076 ,$$

$$\lambda= 0.004 .$$

B86 with modified gradient correction for large density gradients. $$g=-c \left( \rho \left( s \right) \right) ^{4/3}-{\frac {\beta\, \left( \chi \left( s \right) \right) ^{2} \left( \rho \left( s \right) \right) ^{4/3}}{ \left( 1+\lambda\, \left( \chi \left( s \right) \right) ^{2} \right) ^{4/5}}} ,$$

$$G=-c \left( \rho \left( s \right) \right) ^{4/3}-{\frac {\beta\, \left( \chi \left( s \right) \right) ^{2} \left( \rho \left( s \right) \right) ^{4/3}}{ \left( 1+\lambda\, \left( \chi \left( s \right) \right) ^{2} \right) ^{4/5}}} ,$$

$$c=3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$\beta= 0.00375 ,$$

$$\lambda= 0.007 .$$

Re-optimised $\beta$ of B86 used in part 3 of Becke’s 1997 paper. $$g=-{\frac {c \left( \rho \left( s \right) \right) ^{4/3} \left( 1+ \beta\, \left( \chi \left( s \right) \right) ^{2} \right) }{1+\lambda \, \left( \chi \left( s \right) \right) ^{2}}} ,$$

$$G=-{\frac {c \left( \rho \left( s \right) \right) ^{4/3} \left( 1+ \beta\, \left( \chi \left( s \right) \right) ^{2} \right) }{1+\lambda \, \left( \chi \left( s \right) \right) ^{2}}} ,$$

$$c=3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$\beta= 0.00787 ,$$

$$\lambda= 0.004 .$$

$$G=- \left( \rho \left( s \right) \right) ^{4/3} \left( c+{\frac {\beta \, \left( \chi \left( s \right) \right) ^{2}}{1+6\,\beta\,\chi \left( s \right) {\it arcsinh} \left( \chi \left( s \right) \right) }} \right) ,$$

$$g=- \left( \rho \left( s \right) \right) ^{4/3} \left( c+{\frac {\beta \, \left( \chi \left( s \right) \right) ^{2}}{1+6\,\beta\,\chi \left( s \right) {\it arcsinh} \left( \chi \left( s \right) \right) }} \right) ,$$

$$c=3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$\beta= 0.0042 .$$

Correlation functional depending on B86MGC exchange functional with empirical atomic parameters, $t$ and $u$. The exchange functional that is used in conjunction with B88C should replace B88MGC here. $$f=- 0.8\,\rho \left( a \right) \rho \left( b \right) {q}^{2} \left( 1-{ \frac {\ln \left( 1+q \right) }{q}} \right) ,$$

$$q=t \left( x+y \right) ,$$

$$x= 0.5\, \left( c\sqrt [3]{\rho \left( a \right) }+{\frac {\beta\, \left( \chi \left( a \right) \right) ^{2}\sqrt [3]{\rho \left( a \right) }}{ \left( 1+\lambda\, \left( \chi \left( a \right) \right) ^ {2} \right) ^{4/5}}} \right) ^{-1} ,$$

$$y= 0.5\, \left( c\sqrt [3]{\rho \left( b \right) }+{\frac {\beta\, \left( \chi \left( b \right) \right) ^{2}\sqrt [3]{\rho \left( b \right) }}{ \left( 1+\lambda\, \left( \chi \left( b \right) \right) ^ {2} \right) ^{4/5}}} \right) ^{-1} ,$$

$$t= 0.63 ,$$

$$g=- 0.01\,\rho \left( s \right) d{z}^{4} \left( 1-2\,{\frac {\ln \left( 1+1/2\,z \right) }{z}} \right) ,$$

$$z=2\,ur ,$$

$$r= 0.5\,\rho \left( s \right) \left( c \left( \rho \left( s \right) \right) ^{4/3}+{\frac {\beta\, \left( \chi \left( s \right) \right) ^ {2} \left( \rho \left( s \right) \right) ^{4/3}}{ \left( 1+\lambda\, \left( \chi \left( s \right) \right) ^{2} \right) ^{4/5}}} \right) ^{ -1} ,$$

$$u= 0.96 ,$$

$$d=\tau \left( s \right) -1/4\,{\frac {\sigma \left( {\it ss} \right) }{ \rho \left( s \right) }} ,$$

$$G=- 0.01\,\rho \left( s \right) d{z}^{4} \left( 1-2\,{\frac {\ln \left( 1+1/2\,z \right) }{z}} \right) ,$$

$$c=3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$\beta= 0.00375 ,$$

$$\lambda= 0.007 .$$

$\\tau$ dependent Dynamical correlation functional. $$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$f={\frac {E}{1+l \left( \left( \chi \left( a \right) \right) ^{2}+ \left( \chi \left( b \right) \right) ^{2} \right) }} ,$$

$$g={\frac {F\epsilon \left( \rho \left( s \right) ,0 \right) }{H \left( 1+\nu\, \left( \chi \left( s \right) \right) ^{2} \right) ^{2}}} ,$$

$$G={\frac {F\epsilon \left( \rho \left( s \right) ,0 \right) }{H \left( 1+\nu\, \left( \chi \left( s \right) \right) ^{2} \right) ^{2}}} ,$$

$$E=\epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) ,$$

$$l= 0.0031 ,$$

$$F=\tau \left( s \right) -1/4\,{\frac {\sigma \left( {\it ss} \right) }{ \rho \left( s \right) }} ,$$

$$H=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$$\nu= 0.038 ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

This functional needs to be mixed with 0.1943*exact exchange. $$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$A=[ 0.9454, 0.7471,- 4.5961] ,$$

$$B=[ 0.1737, 2.3487,- 2.4868] ,$$

$$C=[ 0.8094, 0.5073, 0.7481] ,$$

$$\lambda=[ 0.006, 0.2, 0.004] ,$$

$$d=1/2\, \left( \chi \left( a \right) \right) ^{2}+1/2\, \left( \chi \left( b \right) \right) ^{2} ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( A_{{0}}+A_{{1 }}\eta \left( d,\lambda_{{1}} \right) +A_{{2}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{2} \right) ,$$

$$\eta \left( \theta,\mu \right) ={\frac {\mu\,\theta}{1+\mu\,\theta}} ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2} \right) -3/8\,\sqrt [ 3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2} \right) ,$$

$$G=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2} \right) -3/8\,\sqrt [ 3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2} \right) ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

Re-parameterization of the B97 functional in a self-consistent procedure by Hamprecht et al. This functional needs to be mixed with 0.21*exact exchange. $$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$A=[ 0.955689, 0.788552,- 5.47869] ,$$

$$B=[ 0.0820011, 2.71681,- 2.87103] ,$$

$$C=[ 0.789518, 0.573805, 0.660975] ,$$

$$\lambda=[ 0.006, 0.2, 0.004] ,$$

$$d=1/2\, \left( \chi \left( a \right) \right) ^{2}+1/2\, \left( \chi \left( b \right) \right) ^{2} ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( A_{{0}}+A_{{1 }}\eta \left( d,\lambda_{{1}} \right) +A_{{2}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{2} \right) ,$$

$$\eta \left( \theta,\mu \right) ={\frac {\mu\,\theta}{1+\mu\,\theta}} ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2} \right) -3/8\,\sqrt [ 3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2} \right) ,$$

$$G=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2} \right) -3/8\,\sqrt [ 3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2} \right) ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

A. D. Becke and M. R. Roussel,Phys. Rev. A 39, 3761 (1989)

$$K=\frac{1}{2}\sum_s \rho_s U_s ,$$ where $$U_s=-(1-e^{-x}-xe^{-x}/2)/b ,$$ $$b=\frac{x^3e^{-x}}{8\pi\rho_s}$$ and $x$ is defined by the nonlinear equation $$\frac{xe^{-2x/3}}{x-2}=\frac{2\pi^{2/3}\rho_s^{5/3}}{3Q_s} ,$$ where $$Q_s=(\upsilon_s-2\gamma D_s)/6 ,$$ $$D_s=\tau_s-\frac{\sigma_{ss}}{4\rho_s}$$ and $$\gamma=1.$$

A. D. Becke and M. R. Roussel,Phys. Rev. A 39, 3761 (1989)

As for BR but with ${\gamma=0.8}$.

Hybrid exchange-correlation functional comprimising Becke’s 1998 exchange and Wigner’s spin-polarised correlation functionals. $$\alpha=-3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$g=\alpha\, \left( \rho \left( s \right) \right) ^{4/3}-{\frac {\beta\, \left( \rho \left( s \right) \right) ^{4/3} \left( \chi \left( s \right) \right) ^{2}}{1+6\,\beta\,\chi \left( s \right) {\it arcsinh} \left( \chi \left( s \right) \right) }} ,$$

$$G=\alpha\, \left( \rho \left( s \right) \right) ^{4/3}-{\frac {\beta\, \left( \rho \left( s \right) \right) ^{4/3} \left( \chi \left( s \right) \right) ^{2}}{1+6\,\beta\,\chi \left( s \right) {\it arcsinh} \left( \chi \left( s \right) \right) }} ,$$

$$f=-4\,c\rho \left( a \right) \rho \left( b \right) {\rho}^{-1} \left( 1 +{\frac {d}{\sqrt [3]{\rho}}} \right) ^{-1} ,$$

$$\beta= 0.0042 ,$$

$$c= 0.04918 ,$$

$$d= 0.349 .$$

R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1974); C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785(1988)

CS1 is formally identical to CS2, except for a reformulation in which the terms involving $\upsilon$ are eliminated by integration by parts. This makes the functional more economical to evaluate. In the limit of exact quadrature, CS1 and CS2 are identical, but small numerical differences appear with finite integration grids.

R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1974); C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785(1988)

CS2 is defined through \begin{aligned} K &=& -a \left({ \rho+2b\rho^{-5/3} \left[ \rho_\alpha t_{\alpha} + \rho_\beta t_{\beta} -\rho t_W \right] e^{-c\rho^{-1/3}} \over 1+d \rho^{-1/3} }\right) \end{aligned} where \begin{aligned} t_{\alpha} &=&\frac{\tau_\alpha}{2}-\frac{\upsilon_\alpha}{8} \\ t_{\beta} &=&\frac{\tau_\beta}{2}-\frac{\upsilon_\beta}{8} \\ t_{W} &=& {1\over 8} {\sigma \over \rho} - {1\over 2} \upsilon \end{aligned} and the constants are $a=0.04918, b=0.132, c=0.2533, d=0.349$.

Automatically generated Slater-Dirac exchange. $$g=-c \left( \rho \left( s \right) \right) ^{4/3} ,$$

$$c=3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} .$$

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).

$$\nonumber \epsilon_c^{\rm SR}(r_s,\zeta,\mu) =\epsilon_c^{\rm PW92}(r_s,\zeta)- \frac{[\phi_2(\zeta)]^3Q\left(\frac{\mu\sqrt{r_s}}{\phi_2(\zeta)}\right)+a_1 \mu^3+a_2 \mu^4+ a_3\mu^5+a_4\mu^6+a_5\mu^8}{(1+b_0^2\mu^2)^4},$$ where $$Q(x)=\frac{2\ln(2)-2}{\pi^2}\ln\left(\frac{1+a\,x+b\,x^2+c\,x^3}{1+a\,x+ d\,x^2}\right),$$ 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 \begin{aligned} a_1 & = & 4 \,b_0^6 \,C_3+b_0^8 \,C_5, \nonumber \\ a_2 & = & 4 \,b_0^6 \,C_2+b_0^8\, C_4+6\, b_0^4 \epsilon_c^{\rm PW92}, \nonumber \\ a_3 & = & b_0^8 \,C_3, \nonumber \\ a_4 & = & b_0^8 \,C_2+4 \,b_0^6\, \epsilon_c^{\rm PW92} \nonumber, \\ a_5 & = & b_0^8\,\epsilon_c^{\rm PW92}, \nonumber\end{aligned} with \begin{aligned} C_2 & = & -\frac{3(1\!-\!\zeta^2)\,g_c(0,r_s,\zeta\!=\!0)}{8\,r_s^3} \nonumber \\ C_3 & = & - (1\!-\!\zeta^2)\frac{g(0,r_s,\zeta\!=\!0)}{\sqrt{2\pi}\, r_s^3} \nonumber \\ C_4 & = & -\frac{9\, c_4(r_s,\zeta)}{64 r_s^3} \nonumber \\ C_5 & = & -\frac{9\, c_5(r_s,\zeta)}{40\sqrt{2 \pi} r_s^3}\nonumber \\ c_4(r_s,\zeta) & = & \left(\frac{1\!+\!\zeta}{2}\right)^2g''\left(0, r_s\left(\frac{2}{1\!+\!\zeta}\right)^{1/3}\!\!\!\!\!\!\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ \left(\frac{1\!-\!\zeta}{2}\right)^2 \times \nonumber \\ & & g''\left(0, r_s\left(\frac{2}{1\!-\!\zeta}\right)^{1/3}\!\!\!\!\!\!\!\!, \,\,\,\,\,\zeta\!=\!1\right) + (1\!-\!\zeta^2)D_2(r_s)-\frac{\phi_8(\zeta)}{5\,\alpha^2\,r_s^2} \nonumber \\ c_5(r_s,\zeta) & = & \left(\frac{1\!+\!\zeta}{2}\right)^2g''\left(0, r_s\left(\frac{2}{1\!+\!\zeta}\right)^{1/3}\!\!\!\!\!\!\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ \left(\frac{1\!-\!\zeta}{2}\right)^2 \times \nonumber \\ & & g''\left(0, r_s\left(\frac{2}{1\!-\!\zeta}\right)^{1/3}\!\!\!\!\!\!\!\!, \,\,\,\,\,\zeta\!=\!1\right)+ (1\!-\!\zeta^2)D_3(r_s),\end{aligned} and \begin{aligned} \phantom{\bigl[} b_0(r_s) = 0.784949\,r_s \\ \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)} \\ \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) \\ \phantom{\Biggl[}D_3(r_s) = \frac{e^{-0.31 r_s}}{r_s^3}\left(-4.95 r_s+ r_s^2\right).\end{aligned} 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 $$g(0,r_s,\zeta\!=\!0)=\frac{1}{2}(1-Br_s+Cr_s^2+Dr_s^3+Er_s^4)\,{\rm e}^{-dr_s},$$ 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 $g_c(0,r_s,\zeta\!=\!0)=g(0,r_s,\zeta\!=\!0)-\frac{1}{2}$.

Toulouse-Colonna-Savin range-separated correlation functional based on PBE, see J. Toulouse et al., J. Chem. Phys. 122, 014110 (2005).

Hartree-Fock exact exchange functional can be used to construct hybrid exchange-correlation functional.

Local-density approximation of exchange energy
for short-range interelectronic interaction ${\rm erf}(\mu r_{12})/r_{12}$,
A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier, Amsterdam, 1996).

$$\epsilon_x^{\rm SR}(r_s,\zeta,\mu) = \frac{3}{4\pi}\frac{\phi_4(\zeta)}{\alpha\,r_s}- \frac{1}{2}(1\!+\!\zeta)^{4/3} f_x\left(r_s,\mu(1\!+\!\zeta)^{-1/3}\right)+\frac{1}{2}(1\!-\!\zeta)^{4/3} f_x\left(r_s,\mu(1\!-\!\zeta)^{-1/3}\right) \nonumber$$ with $$\phi_n(\zeta)=\frac{1}{2}\left[ (1\!+\!\zeta)^{n/3}+(1\!-\!\zeta)^{n/3} \right],$$ $$f_x(r_s,\mu) = -\frac{\mu}{\pi}\biggl[(2y-4y^3)\,e^{-1/4y^2}- 3y+4y^3+ \sqrt{\pi}\,{\rm erf}\left(\frac{1}{2y}\right)\biggr], \qquad y=\frac{\mu\,\alpha\,r_s}{2},$$ and $\alpha=(4/9\pi)^{1/3}$.

Toulouse-Colonna-Savin range-separated exchange functional based on PBE, see J. Toulouse et al., J. Chem. Phys. 122, 014110 (2005).

$$\alpha=-3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} ,$$

$$g= \left( \rho \left( s \right) \right) ^{4/3} \left( \alpha-{\frac {1 }{137}}\, \left( \chi \left( s \right) \right) ^{3/2} \right) ,$$

$$G= \left( \rho \left( s \right) \right) ^{4/3} \left( \alpha-{\frac {1 }{137}}\, \left( \chi \left( s \right) \right) ^{3/2} \right) .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$A=[ 0.51473, 6.9298,- 24.707, 23.110,- 11.323] ,$$

$$B=[ 0.48951,- 0.2607, 0.4329,- 1.9925, 2.4853] ,$$

$$C=[ 1.09163,- 0.7472, 5.0783,- 4.1075, 1.1717] ,$$

$$\lambda=[ 0.006, 0.2, 0.004] ,$$

$$d=1/2\, \left( \chi \left( a \right) \right) ^{2}+1/2\, \left( \chi \left( b \right) \right) ^{2} ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( A_{{0}}+A_{{1 }}\eta \left( d,\lambda_{{1}} \right) +A_{{2}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{2}+A_{{3}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{3}+A_{{4}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{4} \right) ,$$

$$\eta \left( \theta,\mu \right) ={\frac {\mu\,\theta}{1+\mu\,\theta}} ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2}+B_{{3}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{3}+B_{{4}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{4} \right) -3/8 \,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2}+C_{{3}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{3}+C_{{4}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda _{{3}} \right) \right) ^{4} \right) ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$A=[ 0.542352, 7.01464,- 28.3822, 35.0329,- 20.4284] ,$$

$$B=[ 0.562576,- 0.0171436,- 1.30636, 1.05747, 0.885429] ,$$

$$C=[ 1.09025,- 0.799194, 5.57212,- 5.86760, 3.04544] ,$$

$$\lambda=[ 0.006, 0.2, 0.004] ,$$

$$d=1/2\, \left( \chi \left( a \right) \right) ^{2}+1/2\, \left( \chi \left( b \right) \right) ^{2} ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( A_{{0}}+A_{{1 }}\eta \left( d,\lambda_{{1}} \right) +A_{{2}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{2}+A_{{3}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{3}+A_{{4}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{4} \right) ,$$

$$\eta \left( \theta,\mu \right) ={\frac {\mu\,\theta}{1+\mu\,\theta}} ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2}+B_{{3}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{3}+B_{{4}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{4} \right) -3/8 \,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2}+C_{{3}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{3}+C_{{4}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda _{{3}} \right) \right) ^{4} \right) ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) +{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$A=[ 0.72997, 3.35287,- 11.543, 8.08564,- 4.47857] ,$$

$$B=[ 0.222601,- 0.0338622,- 0.012517,- 0.802496, 1.55396] ,$$

$$C=[ 1.0932,- 0.744056, 5.5992,- 6.78549, 4.49357] ,$$

$$\lambda=[ 0.006, 0.2, 0.004] ,$$

$$d=1/2\, \left( \chi \left( a \right) \right) ^{2}+1/2\, \left( \chi \left( b \right) \right) ^{2} ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( A_{{0}}+A_{{1 }}\eta \left( d,\lambda_{{1}} \right) +A_{{2}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{2}+A_{{3}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{3}+A_{{4}} \left( \eta \left( d, \lambda_{{1}} \right) \right) ^{4} \right) ,$$

$$\eta \left( \theta,\mu \right) ={\frac {\mu\,\theta}{1+\mu\,\theta}} ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( B_{{0}}+B_{{ 1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2} } \right) +B_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{2}+B_{{3}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{3}+B_{{4}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{2}} \right) \right) ^{4} \right) -3/8 \,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} \left( C_{{0}}+C_{{1}}\eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) +C_{{2}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{2}+C_{{3}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda_{{3}} \right) \right) ^{3}+C_{{4}} \left( \eta \left( \left( \chi \left( s \right) \right) ^{2},\lambda _{{3}} \right) \right) ^{4} \right) ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

Henderson-Janesko-Scuseria range-separated exchange functional based on a model of an exchange hole derived by a constraint-satisfaction technique, see T. M. Henderson et al., J. Chem. Phys. 128, 194105 (2008).

LSDA exchange functional with density represented as a function of $\tau$. $$g=1/2\,E \left( 2\,\tau \left( s \right) \right) ,$$

$$E \left( \alpha \right) =1/9\,c{5}^{4/5}\sqrt [5]{9} \left( {\frac { \alpha\,\sqrt [3]{3}}{ \left( {\pi }^{2} \right) ^{2/3}}} \right) ^{4/5 } ,$$

$$c=-3/4\,\sqrt [3]{3}\sqrt [3]{{\pi }^{-1}} ,$$

$$G=1/2\,E \left( 2\,\tau \left( s \right) \right) .$$

C. Lee, W. Yang and R. G. Parr, Phys. Rev. B 37, 785(1988); B. Miehlich, A. Savin, H. Stoll and H. Preuss, Chem. Phys. Lett. 157, 200 (1989). $$f=-4\,A\rho \left( a \right) \rho \left( b \right) \left( 1+{\frac {d} {\sqrt [3]{\rho}}} \right) ^{-1}{\rho}^{-1}-AB\omega\, \left( \rho \left( a \right) \rho \left( b \right) \left( 8\,{2}^{2/3}{\it cf}\, \left( \left( \rho \left( a \right) \right) ^{8/3}+ \left( \rho \left( b \right) \right) ^{8/3} \right) + \left( {\frac {47}{18}}-{ \frac {7}{18}}\,\delta \right) \sigma- \left( 5/2-1/18\,\delta \right) \left( \sigma \left( {\it aa} \right) +\sigma \left( {\it bb} \right) \right) -1/9\, \left( \delta-11 \right) \left( {\frac {\rho \left( a \right) \sigma \left( {\it aa} \right) }{\rho}}+{\frac {\rho \left( b \right) \sigma \left( {\it bb} \right) }{\rho}} \right) \right) -2/3 \,{\rho}^{2}\sigma+ \left( 2/3\,{\rho}^{2}- \left( \rho \left( a \right) \right) ^{2} \right) \sigma \left( {\it bb} \right) + \left( 2/3\,{\rho}^{2}- \left( \rho \left( b \right) \right) ^{2} \right) \sigma \left( {\it aa} \right) \right) ,$$

$$\omega={e^{-{\frac {c}{\sqrt [3]{\rho}}}}}{\rho}^{-11/3} \left( 1+{ \frac {d}{\sqrt [3]{\rho}}} \right) ^{-1} ,$$

$$\delta={\frac {c}{\sqrt [3]{\rho}}}+d{\frac {1}{\sqrt [3]{\rho}}} \left( 1+{\frac {d}{\sqrt [3]{\rho}}} \right) ^{-1} ,$$

$${\it cf}=3/10\,{3}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$$A= 0.04918 ,$$

$$B= 0.132 ,$$

$$c= 0.2533 ,$$

$$d= 0.349 .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) ,$$

$${\it ds}=2\,{\it tausMFM}-1/4\,{\frac {\sigma \left( {\it ss} \right) } {\rho \left( s \right) }} ,$$

$${\it Gab} \left( {\it chia},{\it chib} \right) =\sum _{i=0}^{n}{\it cCab}_{{i}} \left( {\frac {{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib}}^{2} \right) }{1+{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib} }^{2} \right) }} \right) ^{i} ,$$

$${\it Gss} \left( {\it chis} \right) =\sum _{i=0}^{n}{\it cCss}_{{i}} \left( {\frac {{\it yCss}\,{{\it chis}}^{2}}{1+{\it yCss}\,{{\it chis} }^{2}}} \right) ^{i} ,$$

$$n=4 ,$$

$${\it cCab}=[ 1.0, 1.09297,- 3.79171, 2.82810,- 10.58909] ,$$

$${\it cCss}=[ 1.0,- 3.05430, 7.61854, 1.47665,- 11.92365] ,$$

$${\it yCab}= 0.0031 ,$$

$${\it yCss}= 0.06 ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) {\it Gab} \left( \chi \left( a \right) ,\chi \left( b \right) \right) ,$$

$$g=1/2\,{\frac {\epsilon \left( \rho \left( s \right) ,0 \right) {\it Gss} \left( \chi \left( s \right) \right) {\it ds}}{{\it tausMFM}}} ,$$

$$G=1/2\,{\frac {\epsilon \left( \rho \left( s \right) ,0 \right) {\it Gss} \left( \chi \left( s \right) \right) {\it ds}}{{\it tausMFM}}} .$$

$$g=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }} ,$$

$$G=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 ,$$

$$n=11 ,$$

$$A=[ 1.0,- 0.56833,- 1.30057, 5.50070, 9.06402,- 32.21075,- 23.73298, 70.22996, 29.88614,- 60.25778,- 13.22205, 15.23694] ,$$

$${\it Fs} \left( {\it ws} \right) =\sum _{i=0}^{n}A_{{i}}{{\it ws}}^{i} ,$$

$${\it ws}={\frac {{\it ts}-1}{{\it ts}+1}} ,$$

$${\it ts}={\frac {{\it tslsda}}{{\it tausMFM}}} ,$$

$${\it tslsda}=3/10\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) ,$$

$${\it ds}=2\,{\it tausMFM}-1/4\,{\frac {\sigma \left( {\it ss} \right) } {\rho \left( s \right) }} ,$$

$${\it Gab} \left( {\it chia},{\it chib} \right) =\sum _{i=0}^{n}{\it cCab}_{{i}} \left( {\frac {{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib}}^{2} \right) }{1+{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib} }^{2} \right) }} \right) ^{i} ,$$

$${\it Gss} \left( {\it chis} \right) =\sum _{i=0}^{n}{\it cCss}_{{i}} \left( {\frac {{\it yCss}\,{{\it chis}}^{2}}{1+{\it yCss}\,{{\it chis} }^{2}}} \right) ^{i} ,$$

$$n=4 ,$$

$${\it cCab}=[ 1.0, 3.78569,- 14.15261,- 7.46589, 17.94491] ,$$

$${\it cCss}=[ 1.0, 3.77344,- 26.04463, 30.69913,- 9.22695] ,$$

$${\it yCab}= 0.0031 ,$$

$${\it yCss}= 0.06 ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) {\it Gab} \left( \chi \left( a \right) ,\chi \left( b \right) \right) ,$$

$$g=1/2\,{\frac {\epsilon \left( \rho \left( s \right) ,0 \right) {\it Gss} \left( \chi \left( s \right) \right) {\it ds}}{{\it tausMFM}}} ,$$

$$G=1/2\,{\frac {\epsilon \left( \rho \left( s \right) ,0 \right) {\it Gss} \left( \chi \left( s \right) \right) {\it ds}}{{\it tausMFM}}} .$$

$$g=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }} ,$$

$$G=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 ,$$

$$n=11 ,$$

$$A=[ 1.0, 0.08151,- 0.43956,- 3.22422, 2.01819, 8.79431,- 0.00295, 9.82029,- 4.82351,- 48.17574, 3.64802, 34.02248] ,$$

$${\it Fs} \left( {\it ws} \right) =\sum _{i=0}^{n}A_{{i}}{{\it ws}}^{i} ,$$

$${\it ws}={\frac {{\it ts}-1}{{\it ts}+1}} ,$$

$${\it ts}={\frac {{\it tslsda}}{{\it tausMFM}}} ,$$

$${\it tslsda}=3/10\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$${\it Gab} \left( {\it chia},{\it chib} \right) =\sum _{i=0}^{n}{\it cCab}_{{i}} \left( {\frac {{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib}}^{2} \right) }{1+{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib} }^{2} \right) }} \right) ^{i} ,$$

$${\it Gss} \left( {\it chis} \right) =\sum _{i=0}^{n}{\it cCss}_{{i}} \left( {\frac {{\it yCss}\,{{\it chis}}^{2}}{1+{\it yCss}\,{{\it chis} }^{2}}} \right) ^{i} ,$$

$$n=4 ,$$

$${\it cCab}=[ 0.8833596, 33.57972,- 70.43548, 49.78271,- 18.52891] ,$$

$${\it cCss}=[ 0.3097855,- 5.528642, 13.47420,- 32.13623, 28.46742] ,$$

$${\it yCab}= 0.0031 ,$$

$${\it yCss}= 0.06 ,$$

$$x=\sqrt { \left( \chi \left( a \right) \right) ^{2}+ \left( \chi \left( b \right) \right) ^{2}} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) ,$$

$${\it tauaMFM}=1/2\,\tau \left( a \right) ,$$

$${\it taubMFM}=1/2\,\tau \left( b \right) ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$z=2\,{\frac {{\it tauaMFM}}{ \left( \rho \left( a \right) \right) ^{5/ 3}}}+2\,{\frac {{\it taubMFM}}{ \left( \rho \left( b \right) \right) ^ {5/3}}}-2\,{\it cf} ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it ds}=1-{\frac { \left( \chi \left( s \right) \right) ^{2}}{4\,{ \it zs}+4\,{\it cf}}} ,$$

$$h \left( x,z,{\it d0},{\it d1},{\it d2},{\it d3},{\it d4},{\it d5}, \alpha \right) ={\frac {{\it d0}}{\lambda \left( x,z,\alpha \right) }}+ {\frac {{\it d1}\,{x}^{2}+{\it d2}\,z}{ \left( \lambda \left( x,z, \alpha \right) \right) ^{2}}}+{\frac {{\it d3}\,{x}^{4}+{\it d4}\,{x}^ {2}z+{\it d5}\,{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3}}} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it dCab}=[ 0.1166404,- 0.09120847,- 0.06726189, 0.00006720580, 0.0008448011, 0.0] ,$$

$${\it dCss}=[ 0.6902145, 0.09847204, 0.2214797,- 0.001968264,- 0.006775479, 0.0] ,$$

$${\it aCab}= 0.003050 ,$$

$${\it aCss}= 0.005151 ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( {\it Gab} \left( \chi \left( a \right) ,\chi \left( b \right) \right) +h \left( x,z,{\it dCab}_{{0}},{\it dCab}_{{1}},{\it dCab}_{{2}},{\it dCab}_{{3}},{\it dCab}_{{4}},{\it dCab}_{{5}},{\it aCab} \right) \right) ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} ,$$

$$G=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} .$$

$$g=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }} ,$$

$$G=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 ,$$

$$n=11 ,$$

$$A=[ 0.4600000,- 0.2206052,- 0.09431788, 2.164494,- 2.556466,- 14.22133, 15.55044, 35.98078,- 27.22754,- 39.24093, 15.22808, 15.22227] ,$$

$${\it Fs} \left( {\it ws} \right) =\sum _{i=0}^{n}A_{{i}}{{\it ws}}^{i} ,$$

$${\it ws}={\frac {{\it ts}-1}{{\it ts}+1}} ,$$

$${\it ts}={\frac {{\it tslsda}}{{\it tausMFM}}} ,$$

$${\it tslsda}=3/10\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$${\it Gab} \left( {\it chia},{\it chib} \right) =\sum _{i=0}^{n}{\it cCab}_{{i}} \left( {\frac {{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib}}^{2} \right) }{1+{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib} }^{2} \right) }} \right) ^{i} ,$$

$${\it Gss} \left( {\it chis} \right) =\sum _{i=0}^{n}{\it cCss}_{{i}} \left( {\frac {{\it yCss}\,{{\it chis}}^{2}}{1+{\it yCss}\,{{\it chis} }^{2}}} \right) ^{i} ,$$

$$n=4 ,$$

$${\it cCab}=[ 3.741593, 218.7098,- 453.1252, 293.6479,- 62.87470] ,$$

$${\it cCss}=[ 0.5094055,- 1.491085, 17.23922,- 38.59018, 28.45044] ,$$

$${\it yCab}= 0.0031 ,$$

$${\it yCss}= 0.06 ,$$

$$x=\sqrt { \left( \chi \left( a \right) \right) ^{2}+ \left( \chi \left( b \right) \right) ^{2}} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) ,$$

$${\it tauaMFM}=1/2\,\tau \left( a \right) ,$$

$${\it taubMFM}=1/2\,\tau \left( b \right) ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$z=2\,{\frac {{\it tauaMFM}}{ \left( \rho \left( a \right) \right) ^{5/ 3}}}+2\,{\frac {{\it taubMFM}}{ \left( \rho \left( b \right) \right) ^ {5/3}}}-2\,{\it cf} ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it ds}=1-{\frac { \left( \chi \left( s \right) \right) ^{2}}{4\,{ \it zs}+4\,{\it cf}}} ,$$

$$h \left( x,z,{\it d0},{\it d1},{\it d2},{\it d3},{\it d4},{\it d5}, \alpha \right) ={\frac {{\it d0}}{\lambda \left( x,z,\alpha \right) }}+ {\frac {{\it d1}\,{x}^{2}+{\it d2}\,z}{ \left( \lambda \left( x,z, \alpha \right) \right) ^{2}}}+{\frac {{\it d3}\,{x}^{4}+{\it d4}\,{x}^ {2}z+{\it d5}\,{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3}}} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it dCab}=[- 2.741539,- 0.6720113,- 0.07932688, 0.001918681,- 0.002032902, 0.0] ,$$

$${\it dCss}=[ 0.4905945,- 0.1437348, 0.2357824, 0.001871015,- 0.003788963, 0.0] ,$$

$${\it aCab}= 0.003050 ,$$

$${\it aCss}= 0.005151 ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( {\it Gab} \left( \chi \left( a \right) ,\chi \left( b \right) \right) +h \left( x,z,{\it dCab}_{{0}},{\it dCab}_{{1}},{\it dCab}_{{2}},{\it dCab}_{{3}},{\it dCab}_{{4}},{\it dCab}_{{5}},{\it aCab} \right) \right) ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} ,$$

$$G=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$${\it Gab} \left( {\it chia},{\it chib} \right) =\sum _{i=0}^{n}{\it cCab}_{{i}} \left( {\frac {{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib}}^{2} \right) }{1+{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib} }^{2} \right) }} \right) ^{i} ,$$

$${\it Gss} \left( {\it chis} \right) =\sum _{i=0}^{n}{\it cCss}_{{i}} \left( {\frac {{\it yCss}\,{{\it chis}}^{2}}{1+{\it yCss}\,{{\it chis} }^{2}}} \right) ^{i} ,$$

$$n=4 ,$$

$${\it cCab}=[ 1.674634, 57.32017, 59.55416,- 231.1007, 125.5199] ,$$

$${\it cCss}=[ 0.1023254,- 2.453783, 29.13180,- 34.94358, 23.15955] ,$$

$${\it yCab}= 0.0031 ,$$

$${\it yCss}= 0.06 ,$$

$$x=\sqrt { \left( \chi \left( a \right) \right) ^{2}+ \left( \chi \left( b \right) \right) ^{2}} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) ,$$

$${\it tauaMFM}=1/2\,\tau \left( a \right) ,$$

$${\it taubMFM}=1/2\,\tau \left( b \right) ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$z=2\,{\frac {{\it tauaMFM}}{ \left( \rho \left( a \right) \right) ^{5/ 3}}}+2\,{\frac {{\it taubMFM}}{ \left( \rho \left( b \right) \right) ^ {5/3}}}-2\,{\it cf} ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it ds}=1-{\frac { \left( \chi \left( s \right) \right) ^{2}}{4\,{ \it zs}+4\,{\it cf}}} ,$$

$$h \left( x,z,{\it d0},{\it d1},{\it d2},{\it d3},{\it d4},{\it d5}, \alpha \right) ={\frac {{\it d0}}{\lambda \left( x,z,\alpha \right) }}+ {\frac {{\it d1}\,{x}^{2}+{\it d2}\,z}{ \left( \lambda \left( x,z, \alpha \right) \right) ^{2}}}+{\frac {{\it d3}\,{x}^{4}+{\it d4}\,{x}^ {2}z+{\it d5}\,{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3}}} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it dCab}=[- 0.6746338,- 0.1534002,- 0.09021521,- 0.001292037,- 0.0002352983, 0.0] ,$$

$${\it dCss}=[ 0.8976746,- 0.2345830, 0.2368173,- 0.0009913890,- 0.01146165, 0.0] ,$$

$${\it aCab}= 0.003050 ,$$

$${\it aCss}= 0.005151 ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( {\it Gab} \left( \chi \left( a \right) ,\chi \left( b \right) \right) +h \left( x,z,{\it dCab}_{{0}},{\it dCab}_{{1}},{\it dCab}_{{2}},{\it dCab}_{{3}},{\it dCab}_{{4}},{\it dCab}_{{5}},{\it aCab} \right) \right) ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} ,$$

$$G=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} .$$

$$g=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }}+{\it eslsda}\,h \left( \chi \left( s \right) ,{\it zs } \right) ,$$

$$G=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }}+{\it eslsda}\,h \left( \chi \left( s \right) ,{\it zs } \right) ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 ,$$

$$n=11 ,$$

$$A=[ 0.1179732,- 1.066708,- 0.1462405, 7.481848, 3.776679,- 44.36118,- 18.30962, 100.3903, 38.64360,- 98.06018,- 25.57716, 35.90404] ,$$

$${\it Fs} \left( {\it ws} \right) =\sum _{i=0}^{n}A_{{i}}{{\it ws}}^{i} ,$$

$${\it ws}={\frac {{\it ts}-1}{{\it ts}+1}} ,$$

$${\it ts}={\frac {{\it tslsda}}{{\it tausMFM}}} ,$$

$${\it tslsda}=3/10\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it eslsda}=-3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} ,$$

$$d=[- 0.1179732,- 0.002500000,- 0.01180065, 0.0, 0.0, 0.0] ,$$

$$\alpha= 0.001867 ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$h \left( x,z \right) ={\frac {d_{{0}}}{\lambda \left( x,z,\alpha \right) }}+{\frac {d_{{1}}{x}^{2}+d_{{2}}z}{ \left( \lambda \left( x,z ,\alpha \right) \right) ^{2}}}+{\frac {d_{{3}}{x}^{4}+d_{{4}}{x}^{2}z+ d_{{5}}{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3} }} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) .$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$${\it Gab} \left( {\it chia},{\it chib} \right) =\sum _{i=0}^{n}{\it cCab}_{{i}} \left( {\frac {{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib}}^{2} \right) }{1+{\it yCab}\, \left( {{\it chia}}^{2}+{{\it chib} }^{2} \right) }} \right) ^{i} ,$$

$${\it Gss} \left( {\it chis} \right) =\sum _{i=0}^{n}{\it cCss}_{{i}} \left( {\frac {{\it yCss}\,{{\it chis}}^{2}}{1+{\it yCss}\,{{\it chis} }^{2}}} \right) ^{i} ,$$

$$n=4 ,$$

$${\it cCab}=[ 0.6042374, 177.6783,- 251.3252, 76.35173,- 12.55699] ,$$

$${\it cCss}=[ 0.5349466, 0.5396620,- 31.61217, 51.49592,- 29.19613] ,$$

$${\it yCab}= 0.0031 ,$$

$${\it yCss}= 0.06 ,$$

$$x=\sqrt { \left( \chi \left( a \right) \right) ^{2}+ \left( \chi \left( b \right) \right) ^{2}} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) ,$$

$${\it tauaMFM}=1/2\,\tau \left( a \right) ,$$

$${\it taubMFM}=1/2\,\tau \left( b \right) ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$z=2\,{\frac {{\it tauaMFM}}{ \left( \rho \left( a \right) \right) ^{5/ 3}}}+2\,{\frac {{\it taubMFM}}{ \left( \rho \left( b \right) \right) ^ {5/3}}}-2\,{\it cf} ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it ds}=1-{\frac { \left( \chi \left( s \right) \right) ^{2}}{4\,{ \it zs}+4\,{\it cf}}} ,$$

$$h \left( x,z,{\it d0},{\it d1},{\it d2},{\it d3},{\it d4},{\it d5}, \alpha \right) ={\frac {{\it d0}}{\lambda \left( x,z,\alpha \right) }}+ {\frac {{\it d1}\,{x}^{2}+{\it d2}\,z}{ \left( \lambda \left( x,z, \alpha \right) \right) ^{2}}}+{\frac {{\it d3}\,{x}^{4}+{\it d4}\,{x}^ {2}z+{\it d5}\,{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3}}} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it dCab}=[ 0.3957626,- 0.5614546, 0.01403963, 0.0009831442,- 0.003577176, 0.0] ,$$

$${\it dCss}=[ 0.4650534, 0.1617589, 0.1833657, 0.0004692100,- 0.004990573, 0.0] ,$$

$${\it aCab}= 0.003050 ,$$

$${\it aCss}= 0.005151 ,$$

$$f= \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) \left( {\it Gab} \left( \chi \left( a \right) ,\chi \left( b \right) \right) +h \left( x,z,{\it dCab}_{{0}},{\it dCab}_{{1}},{\it dCab}_{{2}},{\it dCab}_{{3}},{\it dCab}_{{4}},{\it dCab}_{{5}},{\it aCab} \right) \right) ,$$

$$g=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} ,$$

$$G=\epsilon \left( \rho \left( s \right) ,0 \right) \left( {\it Gss} \left( \chi \left( s \right) \right) +h \left( \chi \left( s \right) ,{\it zs},{\it dCss}_{{0}},{\it dCss}_{{1}},{\it dCss}_{{2}},{\it dCss} _{{3}},{\it dCss}_{{4}},{\it dCss}_{{5}},{\it aCss} \right) \right) { \it ds} .$$

$$g=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }}+{\it eslsda}\,h \left( \chi \left( s \right) ,{\it zs } \right) ,$$

$$G=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }}+{\it eslsda}\,h \left( \chi \left( s \right) ,{\it zs } \right) ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 ,$$

$$n=11 ,$$

$$A=[ 0.3987756, 0.2548219, 0.3923994,- 2.103655,- 6.302147, 10.97615, 30.97273,- 23.18489,- 56.73480, 21.60364, 34.21814,- 9.049762] ,$$

$${\it Fs} \left( {\it ws} \right) =\sum _{i=0}^{n}A_{{i}}{{\it ws}}^{i} ,$$

$${\it ws}={\frac {{\it ts}-1}{{\it ts}+1}} ,$$

$${\it ts}={\frac {{\it tslsda}}{{\it tausMFM}}} ,$$

$${\it tslsda}=3/10\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it eslsda}=-3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} ,$$

$$d=[ 0.6012244, 0.004748822,- 0.008635108,- 0.000009308062, 0.00004482811, 0.0] ,$$

$$\alpha= 0.001867 ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$h \left( x,z \right) ={\frac {d_{{0}}}{\lambda \left( x,z,\alpha \right) }}+{\frac {d_{{1}}{x}^{2}+d_{{2}}z}{ \left( \lambda \left( x,z ,\alpha \right) \right) ^{2}}}+{\frac {d_{{3}}{x}^{4}+d_{{4}}{x}^{2}z+ d_{{5}}{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3} }} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) .$$

$$g=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }}+{\it eslsda}\,h \left( \chi \left( s \right) ,{\it zs } \right) ,$$

$$G=-3/4\,{\frac {\sqrt [3]{6}\sqrt [3]{{\pi }^{2}} \left( \rho \left( s \right) \right) ^{4/3}F \left( S \right) {\it Fs} \left( {\it ws} \right) }{\pi }}+{\it eslsda}\,h \left( \chi \left( s \right) ,{\it zs } \right) ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 ,$$

$$n=11 ,$$

$$A=[ 0.5877943,- 0.1371776, 0.2682367,- 2.515898,- 2.978892, 8.710679, 16.88195,- 4.489724,- 32.99983,- 14.49050, 20.43747, 12.56504] ,$$

$${\it Fs} \left( {\it ws} \right) =\sum _{i=0}^{n}A_{{i}}{{\it ws}}^{i} ,$$

$${\it ws}={\frac {{\it ts}-1}{{\it ts}+1}} ,$$

$${\it ts}={\frac {{\it tslsda}}{{\it tausMFM}}} ,$$

$${\it tslsda}=3/10\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it eslsda}=-3/8\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\pi }^{-1}} \left( \rho \left( s \right) \right) ^{4/3} ,$$

$$d=[ 0.1422057, 0.0007370319,- 0.01601373, 0.0, 0.0, 0.0] ,$$

$$\alpha= 0.001867 ,$$

$${\it zs}=2\,{\frac {{\it tausMFM}}{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$h \left( x,z \right) ={\frac {d_{{0}}}{\lambda \left( x,z,\alpha \right) }}+{\frac {d_{{1}}{x}^{2}+d_{{2}}z}{ \left( \lambda \left( x,z ,\alpha \right) \right) ^{2}}}+{\frac {d_{{3}}{x}^{4}+d_{{4}}{x}^{2}z+ d_{{5}}{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3} }} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it cf}=3/5\,{6}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$${\it tausMFM}=1/2\,\tau \left( s \right) .$$

Meta-GGA correlation functional based on first principles, see M. Modrzejewski et al., J. Chem. Phys. 137, 204121 (2012).

$$g=-3\,{\frac {\pi \, \left( \rho \left( s \right) \right) ^{3}}{\tau \left( s \right) -1/4\,\upsilon \left( s \right) }} .$$

MK00 with gradient correction of the form of B88X but with different empirical parameter. $$g=-3\,{\frac {\pi \, \left( \rho \left( s \right) \right) ^{3}}{\tau \left( s \right) -1/4\,\upsilon \left( s \right) }}-{\frac {\beta\, \left( \rho \left( s \right) \right) ^{4/3} \left( \chi \left( s \right) \right) ^{2}}{1+6\,\beta\,\chi \left( s \right) {\it arcsinh} \left( \chi \left( s \right) \right) }} ,$$

$$\beta= 0.0016 ,$$

$$G=-3\,{\frac {\pi \, \left( \rho \left( s \right) \right) ^{3}}{\tau \left( s \right) -1/4\,\upsilon \left( s \right) }}-{\frac {\beta\, \left( \rho \left( s \right) \right) ^{4/3} \left( \chi \left( s \right) \right) ^{2}}{1+6\,\beta\,\chi \left( s \right) {\it arcsinh} \left( \chi \left( s \right) \right) }} .$$

Gradient correction to VWN. $$f=\rho\,e+{\frac {{e^{-\Phi}}C \left( r \right) \sigma}{d{\rho}^{4/3}}} ,$$

$$r=1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{\frac {1}{\pi \,\rho}}} ,$$

$$x=\sqrt {r} ,$$

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

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

$$l=[- 0.10498,- 0.325,- 0.0047584] ,$$

$$m=[ 3.72744, 7.06042, 1.13107] ,$$

$$n=[ 12.9352, 18.0578, 13.0045] ,$$

$$e=\Lambda+\omega\,y \left( 1+h{\zeta}^{4} \right) ,$$

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

$$h=4/9\,{\frac {\lambda-\Lambda}{ \left( \sqrt [3]{2}-1 \right) \omega}} -1 ,$$

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

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

$$\omega=q \left( k_{{3}},l_{{3}},m_{{3}},n_{{3}} \right) ,$$

$$q \left( A,p,c,d \right) =A \left( \ln \left( {\frac {{x}^{2}}{X \left( x,c,d \right) }} \right) +2\,c\arctan \left( {\frac {Q \left( c ,d \right) }{2\,x+c}} \right) \left( Q \left( c,d \right) \right) ^{- 1}-cp \left( \ln \left( {\frac { \left( x-p \right) ^{2}}{X \left( x,c ,d \right) }} \right) +2\, \left( c+2\,p \right) \arctan \left( {\frac {Q \left( c,d \right) }{2\,x+c}} \right) \left( Q \left( c,d \right) \right) ^{-1} \right) \left( X \left( p,c,d \right) \right) ^{-1} \right) ,$$

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

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

$$\Phi= 0.007390075\,{\frac {z\sqrt {\sigma}}{C \left( r \right) {\rho}^{ 7/6}}} ,$$

$$d=\sqrt [3]{2}\sqrt { \left( 1/2+1/2\,\zeta \right) ^{5/3}+ \left( 1/2- 1/2\,\zeta \right) ^{5/3}} ,$$

$$C \left( r \right) = 0.001667+{\frac { 0.002568+\alpha\,r+\beta\,{r}^{2 }}{1+\xi\,r+\delta\,{r}^{2}+10000\,\beta\,{r}^{3}}} ,$$

$$z= 0.11 ,$$

$$\alpha= 0.023266 ,$$

$$\beta= 0.000007389 ,$$

$$\xi= 8.723 ,$$

$$\delta= 0.472 .$$

$$f=\rho\, \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) +H \left( d,\rho \left( a \right) ,\rho \left( b \right) \right) \right) ,$$

$$G=\rho\, \left( \epsilon \left( \rho \left( s \right) ,0 \right) +C \left( Q,\rho \left( s \right) ,0 \right) \right) ,$$

$$d=1/12\,{\frac {\sqrt {\sigma}{3}^{5/6}}{u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \sqrt [6]{{\pi }^{-1}}{\rho}^{ 7/6}}} ,$$

$$u \left( \alpha,\beta \right) =1/2\, \left( 1+\zeta \left( \alpha,\beta \right) \right) ^{2/3}+1/2\, \left( 1-\zeta \left( \alpha,\beta \right) \right) ^{2/3} ,$$

$$H \left( d,\alpha,\beta \right) =1/2\, \left( u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \right) ^{3}{\lambda}^{2}\ln \left( 1+2\,{\frac {\iota\, \left( {d}^{2}+A \left( \alpha,\beta \right) {d}^{4} \right) }{\lambda\, \left( 1+A \left( \alpha,\beta \right) {d}^{2}+ \left( A \left( \alpha,\beta \right) \right) ^{2}{d} ^{4} \right) }} \right) {\iota}^{-1} ,$$

$$A \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-2\,{ \frac {\iota\,\epsilon \left( \alpha,\beta \right) }{ \left( u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \right) ^{3}{ \lambda}^{2}}}}}-1 \right) ^{-1} ,$$

$$\iota= 0.0716 ,$$

$$\lambda=\nu\,\kappa ,$$

$$\nu=16\,{\frac {\sqrt [3]{3}\sqrt [3]{{\pi }^{2}}}{\pi }} ,$$

$$\kappa= 0.004235 ,$$

$$Z=- 0.001667 ,$$

$$\phi \left( r \right) =\theta \left( r \right) -Z ,$$

$$\theta \left( r \right) ={\frac {1}{1000}}\,{\frac { 2.568+\Xi\,r+\Phi \,{r}^{2}}{1+\Lambda\,r+\Upsilon\,{r}^{2}+10\,\Phi\,{r}^{3}}} ,$$

$$\Xi= 23.266 ,$$

$$\Phi= 0.007389 ,$$

$$\Lambda= 8.723 ,$$

$$\Upsilon= 0.472 ,$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) =e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3 }},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P_{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^{4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2}},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y _{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}} ,U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{4} ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$$C \left( d,\alpha,\beta \right) =K \left( Q,\alpha,\beta \right) +M \left( Q,\alpha,\beta \right) ,$$

$$M \left( d,\alpha,\beta \right) = 0.5\,\nu\, \left( \phi \left( r \left( \alpha,\beta \right) \right) -\kappa-3/7\,Z \right) {d}^{2}{e^ {- 335.9789467\,{\frac {{3}^{2/3}{d}^{2}}{\sqrt [3]{{\pi }^{5}\rho}}}}} ,$$

$$K \left( d,\alpha,\beta \right) = 0.2500000000\,{\lambda}^{2}\ln \left( 1+2\,{\frac {\iota\, \left( {d}^{2}+N \left( \alpha,\beta \right) {d}^{4} \right) }{\lambda\, \left( 1+N \left( \alpha,\beta \right) {d}^{2}+ \left( N \left( \alpha,\beta \right) \right) ^{2}{d} ^{4} \right) }} \right) {\iota}^{-1} ,$$

$$N \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-4\,{ \frac {\iota\,\epsilon \left( \alpha,\beta \right) }{{\lambda}^{2}}}}}- 1 \right) ^{-1} ,$$

$$Q=1/12\,{\frac {\sqrt {\sigma \left( {\it ss} \right) }\sqrt [3]{2}{3}^ {5/6}}{\sqrt [6]{{\pi }^{-1}}{\rho}^{7/6}}} .$$

$$g=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$G=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$E \left( n \right) =-3/4\,{\frac {\sqrt [3]{3}\sqrt [3]{{\pi }^{2}}{n}^ {4/3}F \left( S \right) }{\pi }} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 0.804 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 .$$

Changes the value of the constant R from the original PBEX functional $$g=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$G=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$E \left( n \right) =-3/4\,{\frac {\sqrt [3]{3}\sqrt [3]{{\pi }^{2}}{n}^ {4/3}F \left( S \right) }{\pi }} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) =1+R-R \left( 1+{\frac {\mu\,{S}^{2}}{R}} \right) ^{ -1} ,$$

$$R= 1.245 ,$$

$$\mu=1/3\,\delta\,{\pi }^{2} ,$$

$$\delta= 0.066725 .$$

GGA Exchange Functional. $$g=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$E \left( n \right) =-3/4\,\sqrt [3]{3}\sqrt [3]{{\pi }^{-1}}{n}^{4/3}F \left( S \right) ,$$

$$F \left( S \right) = \left( 1+ 1.296\,{S}^{2}+14\,{S}^{4}+ 0.2\,{S}^{6} \right) ^{1/15} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$G=1/2\,E \left( 2\,\rho \left( s \right) \right) .$$

$$f=\rho\, \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) +H \left( d,\rho \left( a \right) ,\rho \left( b \right) \right) \right) ,$$

$$G=\rho\, \left( \epsilon \left( \rho \left( s \right) ,0 \right) +C \left( Q,\rho \left( s \right) ,0 \right) \right) ,$$

$$d=1/12\,{\frac {\sqrt {\sigma}{3}^{5/6}}{u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \sqrt [6]{{\pi }^{-1}}{\rho}^{ 7/6}}} ,$$

$$u \left( \alpha,\beta \right) =1/2\, \left( 1+\zeta \left( \alpha,\beta \right) \right) ^{2/3}+1/2\, \left( 1-\zeta \left( \alpha,\beta \right) \right) ^{2/3} ,$$

$$H \left( d,\alpha,\beta \right) =L \left( d,\alpha,\beta \right) +J \left( d,\alpha,\beta \right) ,$$

$$L \left( d,\alpha,\beta \right) =1/2\, \left( u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \right) ^{3}{\lambda}^{2}\ln \left( 1+2\,{\frac {\iota\, \left( {d}^{2}+A \left( \alpha,\beta \right) {d}^{4} \right) }{\lambda\, \left( 1+A \left( \alpha,\beta \right) {d}^{2}+ \left( A \left( \alpha,\beta \right) \right) ^{2}{d} ^{4} \right) }} \right) {\iota}^{-1} ,$$

$$J \left( d,\alpha,\beta \right) =\nu\, \left( \phi \left( r \left( \alpha,\beta \right) \right) -\kappa-3/7\,Z \right) \left( u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \right) ^{3}{d}^ {2}{e^{-{\frac {400}{3}}\,{\frac { \left( u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \right) ^{4}{3}^{2/3}{d}^{2}} {\sqrt [3]{{\pi }^{5}\rho}}}}} ,$$

$$A \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-2\,{ \frac {\iota\,\epsilon \left( \alpha,\beta \right) }{ \left( u \left( \rho \left( a \right) ,\rho \left( b \right) \right) \right) ^{3}{ \lambda}^{2}}}}}-1 \right) ^{-1} ,$$

$$\iota= 0.09 ,$$

$$\lambda=\nu\,\kappa ,$$

$$\nu=16\,{\frac {\sqrt [3]{3}\sqrt [3]{{\pi }^{2}}}{\pi }} ,$$

$$\kappa= 0.004235 ,$$

$$Z=- 0.001667 ,$$

$$\phi \left( r \right) =\theta \left( r \right) -Z ,$$

$$\theta \left( r \right) ={\frac {1}{1000}}\,{\frac { 2.568+\Xi\,r+\Phi \,{r}^{2}}{1+\Lambda\,r+\Upsilon\,{r}^{2}+10\,\Phi\,{r}^{3}}} ,$$

$$\Xi= 23.266 ,$$

$$\Phi= 0.007389 ,$$

$$\Lambda= 8.723 ,$$

$$\Upsilon= 0.472 ,$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) =e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3 }},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P_{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^{4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2}},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y _{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}} ,U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{4} ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 ,$$

$$C \left( d,\alpha,\beta \right) =K \left( Q,\alpha,\beta \right) +M \left( Q,\alpha,\beta \right) ,$$

$$M \left( d,\alpha,\beta \right) = 0.5\,\nu\, \left( \phi \left( r \left( \alpha,\beta \right) \right) -\kappa-3/7\,Z \right) {d}^{2}{e^ {- 335.9789467\,{\frac {{3}^{2/3}{d}^{2}}{\sqrt [3]{{\pi }^{5}\rho}}}}} ,$$

$$K \left( d,\alpha,\beta \right) = 0.2500000000\,{\lambda}^{2}\ln \left( 1+2\,{\frac {\iota\, \left( {d}^{2}+N \left( \alpha,\beta \right) {d}^{4} \right) }{\lambda\, \left( 1+N \left( \alpha,\beta \right) {d}^{2}+ \left( N \left( \alpha,\beta \right) \right) ^{2}{d} ^{4} \right) }} \right) {\iota}^{-1} ,$$

$$N \left( \alpha,\beta \right) =2\,\iota{\lambda}^{-1} \left( {e^{-4\,{ \frac {\iota\,\epsilon \left( \alpha,\beta \right) }{{\lambda}^{2}}}}}- 1 \right) ^{-1} ,$$

$$Q=1/12\,{\frac {\sqrt {\sigma \left( {\it ss} \right) }\sqrt [3]{2}{3}^ {5/6}}{\sqrt [6]{{\pi }^{-1}}{\rho}^{7/6}}} .$$

$$g=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$G=1/2\,E \left( 2\,\rho \left( s \right) \right) ,$$

$$E \left( n \right) =-3/4\,{\frac {\sqrt [3]{3}\sqrt [3]{{\pi }^{2}}{n}^ {4/3}F \left( S \right) }{\pi }} ,$$

$$S=1/12\,{\frac {\chi \left( s \right) {6}^{2/3}}{\sqrt [3]{{\pi }^{2}}} } ,$$

$$F \left( S \right) ={\frac {1+ 0.19645\,S{\it arcsinh} \left( 7.7956\, S \right) + \left( 0.2743- 0.1508\,{e^{-100\,{S}^{2}}} \right) {S}^{2} }{1+ 0.19645\,S{\it arcsinh} \left( 7.7956\,S \right) + 0.004\,{S}^{4} }} .$$

Electron-gas correlation energy. $$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$f=\rho\,\epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) ,$$

$$\epsilon \left( \alpha,\beta \right) =e \left( r \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( r \left( \alpha,\beta \right) ,T_{{3}},U_{{3 }},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P_{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^{4} \right) }{c}}+ \left( e \left( r \left( \alpha,\beta \right) ,T_{{2}},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y _{{2}},P_{{2}} \right) -e \left( r \left( \alpha,\beta \right) ,T_{{1}} ,U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{4} ,$$

$$r \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

$$g=\rho \left( s \right) .$$

Automatically generated Thomas-Fermi Kinetic Energy. $$g={\it ctf}\, \left( \rho \left( s \right) \right) ^{5/3} ,$$

$${\it ctf}=3/10\,{2}^{2/3}{3}^{2/3} \left( {\pi }^{2} \right) ^{2/3} .$$

Density and gradient dependent first row exchange-correlation functional. $$t=[7/6,4/3,3/2,5/3,4/3,3/2,5/3,{\frac {11}{6}},3/2,5/3,{\frac {11}{6}}, 2,3/2,5/3,{\frac {11}{6}},2,7/6,4/3,3/2,5/3,1] ,$$

$$u=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0] ,$$

$$v=[0,0,0,0,1,1,1,1,2,2,2,2,0,0,0,0,0,0,0,0,0] ,$$

$$w=[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0] ,$$

$$\omega=[- 0.728255, 0.331699,- 1.02946, 0.235703,- 0.0876221, 0.140854, 0.0336982,- 0.0353615, 0.00497930,- 0.0645900, 0.0461795,- 0.00757191, - 0.00242717, 0.0428140,- 0.0744891, 0.0386577,- 0.352519, 2.19805,- 3.72927, 1.94441, 0.128877] ,$$

$$n=21 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$S_{{i}}= \left( {\frac {\rho \left( a \right) -\rho \left( b \right) }{ \rho}} \right) ^{2\,u_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$Y_{{i}}= \left( {\frac {\sigma \left( {\it aa} \right) +\sigma \left( { \it bb} \right) -2\,\sqrt {\sigma \left( {\it aa} \right) }\sqrt { \sigma \left( {\it bb} \right) }}{{\rho}^{8/3}}} \right) ^{w_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}S_{{i}}X_{{i}}Y_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}} \left( {\frac {\sigma \left( {\it ss} \right) }{ \left( \rho \left( s \right) \right) ^{8/3}}} \right) ^{w_{{i}}} \left( \left( \rho \left( s \right) \right) ^{4/3\,v_{{i}}} \right) ^{-1} .$$

Density and gradient dependent first row exchange-correlation functional. $$t=[{\frac {13}{12}},7/6,4/3,3/2,5/3,{\frac {17}{12}},3/2,5/3,{\frac {11 }{6}},5/3,{\frac {11}{6}},2,5/3,{\frac {11}{6}},2,7/6,4/3,3/2,5/3] ,$$

$$u=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1] ,$$

$$v=[0,0,0,0,0,1,1,1,1,2,2,2,0,0,0,0,0,0,0] ,$$

$$w=[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0] ,$$

$$\omega=[ 0.678831,- 1.75821, 1.27676,- 1.60789, 0.365610,- 0.181327, 0.146973, 0.147141,- 0.0716917,- 0.0407167, 0.0214625,- 0.000768156, 0.0310377,- 0.0720326, 0.0446562,- 0.266802, 1.50822,- 1.94515, 0.679078] ,$$

$$n=19 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$S_{{i}}= \left( {\frac {\rho \left( a \right) -\rho \left( b \right) }{ \rho}} \right) ^{2\,u_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$Y_{{i}}= \left( {\frac {\sigma \left( {\it aa} \right) +\sigma \left( { \it bb} \right) -2\,\sqrt {\sigma \left( {\it aa} \right) }\sqrt { \sigma \left( {\it bb} \right) }}{{\rho}^{8/3}}} \right) ^{w_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}S_{{i}}X_{{i}}Y_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}} \left( {\frac {\sigma \left( {\it ss} \right) }{ \left( \rho \left( s \right) \right) ^{8/3}}} \right) ^{w_{{i}}} \left( \left( \rho \left( s \right) \right) ^{4/3\,v_{{i}}} \right) ^{-1} .$$

Density and gradient dependent first and second row exchange-correlation functional. $$t=[7/6,4/3,3/2,5/3,{\frac {17}{12}},3/2,5/3,{\frac {11}{6}},5/3,{\frac {11}{6}},2,5/3,{\frac {11}{6}},2,7/6,4/3,3/2,5/3,{\frac {13}{12}}] ,$$

$$u=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0] ,$$

$$v=[0,0,0,0,1,1,1,1,2,2,2,0,0,0,0,0,0,0,0] ,$$

$$w=[0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0] ,$$

$$\omega=[- 0.142542,- 0.783603,- 0.188875, 0.0426830,- 0.304953, 0.430407,- 0.0997699, 0.00355789,- 0.0344374, 0.0192108,- 0.00230906, 0.0235189,- 0.0331157, 0.0121316, 0.441190,- 2.27167, 4.03051,- 2.28074, 0.0360204] ,$$

$$n=19 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$S_{{i}}= \left( {\frac {\rho \left( a \right) -\rho \left( b \right) }{ \rho}} \right) ^{2\,u_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$Y_{{i}}= \left( {\frac {\sigma \left( {\it aa} \right) +\sigma \left( { \it bb} \right) -2\,\sqrt {\sigma \left( {\it aa} \right) }\sqrt { \sigma \left( {\it bb} \right) }}{{\rho}^{8/3}}} \right) ^{w_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}S_{{i}}X_{{i}}Y_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}} \left( {\frac {\sigma \left( {\it ss} \right) }{ \left( \rho \left( s \right) \right) ^{8/3}}} \right) ^{w_{{i}}} \left( \left( \rho \left( s \right) \right) ^{4/3\,v_{{i}}} \right) ^{-1} .$$

Density an gradient dependent first and second row exchange-correlation functional. $$t=[7/6,4/3,3/2,5/3,{\frac {17}{12}},3/2,5/3,{\frac {11}{6}},5/3,{\frac {11}{6}},2,5/3,{\frac {11}{6}},2,7/6,4/3,3/2,5/3,{\frac {13}{12}}] ,$$

$$u=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0] ,$$

$$v=[0,0,0,0,1,1,1,1,2,2,2,0,0,0,0,0,0,0,0] ,$$

$$w=[0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0] ,$$

$$\omega=[ 0.0677353,- 1.06763,- 0.0419018, 0.0226313,- 0.222478, 0.283432,- 0.0165089,- 0.0167204,- 0.0332362, 0.0162254,- 0.000984119, 0.0376713,- 0.0653419, 0.0222835, 0.375782,- 1.90675, 3.22494,- 1.68698,- 0.0235810] ,$$

$$n=19 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$S_{{i}}= \left( {\frac {\rho \left( a \right) -\rho \left( b \right) }{ \rho}} \right) ^{2\,u_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$Y_{{i}}= \left( {\frac {\sigma \left( {\it aa} \right) +\sigma \left( { \it bb} \right) -2\,\sqrt {\sigma \left( {\it aa} \right) }\sqrt { \sigma \left( {\it bb} \right) }}{{\rho}^{8/3}}} \right) ^{w_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}S_{{i}}X_{{i}}Y_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}} \left( {\frac {\sigma \left( {\it ss} \right) }{ \left( \rho \left( s \right) \right) ^{8/3}}} \right) ^{w_{{i}}} \left( \left( \rho \left( s \right) \right) ^{4/3\,v_{{i}}} \right) ^{-1} .$$

Density and gradient dependent first row exchange-correlation functional for closed shell systems. Total energies are improved by adding $DN$, where $N$ is the number of electrons and $D=0.1863$. $$t=[7/6,4/3,3/2,5/3,4/3,3/2,5/3,{\frac {11}{6}},3/2,5/3,{\frac {11}{6}}, 2] ,$$

$$v=[0,0,0,0,1,1,1,1,2,2,2,2] ,$$

$$\omega=[- 0.864448, 0.565130,- 1.27306, 0.309681,- 0.287658, 0.588767,- 0.252700, 0.0223563, 0.0140131,- 0.0826608, 0.0556080,- 0.00936227] ,$$

$$n=12 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}X_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,{\frac {\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} .$$

Density and gradient dependent first row exchange-correlation functional. FCFO = FC + open shell fitting. $$t=[7/6,4/3,3/2,5/3,4/3,3/2,5/3,{\frac {11}{6}},3/2,5/3,{\frac {11}{6}}, 2,3/2,5/3,{\frac {11}{6}},2,7/6,4/3,3/2,5/3] ,$$

$$u=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1] ,$$

$$v=[0,0,0,0,1,1,1,1,2,2,2,2,0,0,0,0,0,0,0,0] ,$$

$$w=[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0] ,$$

$$\omega=[- 0.864448, 0.565130,- 1.27306, 0.309681,- 0.287658, 0.588767,- 0.252700, 0.0223563, 0.0140131,- 0.0826608, 0.0556080,- 0.00936227,- 0.00677146, 0.0515199,- 0.0874213, 0.0423827, 0.431940,- 0.691153,- 0.637866, 1.07565] ,$$

$$n=20 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$S_{{i}}= \left( {\frac {\rho \left( a \right) -\rho \left( b \right) }{ \rho}} \right) ^{2\,u_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$Y_{{i}}= \left( {\frac {\sigma \left( {\it aa} \right) +\sigma \left( { \it bb} \right) -2\,\sqrt {\sigma \left( {\it aa} \right) }\sqrt { \sigma \left( {\it bb} \right) }}{{\rho}^{8/3}}} \right) ^{w_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}S_{{i}}X_{{i}}Y_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}} \left( {\frac {\sigma \left( {\it ss} \right) }{ \left( \rho \left( s \right) \right) ^{8/3}}} \right) ^{w_{{i}}} \left( \left( \rho \left( s \right) \right) ^{4/3\,v_{{i}}} \right) ^{-1} .$$

Density and gradient dependent first row exchange-correlation functional. $$t=[7/6,4/3,3/2,5/3,4/3,3/2,5/3,{\frac {11}{6}},3/2,5/3,{\frac {11}{6}}, 2,3/2,5/3,{\frac {11}{6}},2,7/6,4/3,3/2,5/3] ,$$

$$u=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1] ,$$

$$v=[0,0,0,0,1,1,1,1,2,2,2,2,0,0,0,0,0,0,0,0] ,$$

$$w=[0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0] ,$$

$$\omega=[- 0.962998, 0.860233,- 1.54092, 0.381602,- 0.210208, 0.391496,- 0.107660,- 0.0105324, 0.00837384,- 0.0617859, 0.0383072,- 0.00526905,- 0.00381514, 0.0321541,- 0.0568280, 0.0288585, 0.368326,- 0.328799,- 1.22595, 1.36412] ,$$

$$n=20 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$S_{{i}}= \left( {\frac {\rho \left( a \right) -\rho \left( b \right) }{ \rho}} \right) ^{2\,u_{{i}}} ,$$

$$X_{{i}}=1/2\,{\frac { \left( \sqrt {\sigma \left( {\it aa} \right) } \right) ^{v_{{i}}}+ \left( \sqrt {\sigma \left( {\it bb} \right) } \right) ^{v_{{i}}}}{{\rho}^{4/3\,v_{{i}}}}} ,$$

$$Y_{{i}}= \left( {\frac {\sigma \left( {\it aa} \right) +\sigma \left( { \it bb} \right) -2\,\sqrt {\sigma \left( {\it aa} \right) }\sqrt { \sigma \left( {\it bb} \right) }}{{\rho}^{8/3}}} \right) ^{w_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}}S_{{i}}X_{{i}}Y_{{i}} ,$$

$$G=\sum _{i=1}^{n}1/2\,\omega_{{i}} \left( \rho \left( s \right) \right) ^{t_{{i}}} \left( \sqrt {\sigma \left( {\it ss} \right) } \right) ^{v_{{i}}} \left( {\frac {\sigma \left( {\it ss} \right) }{ \left( \rho \left( s \right) \right) ^{8/3}}} \right) ^{w_{{i}}} \left( \left( \rho \left( s \right) \right) ^{4/3\,v_{{i}}} \right) ^{-1} .$$

Density dependent first row exchange-correlation functional for closed shell systems. $$t=[7/6,4/3,3/2,5/3] ,$$

$$\omega=[- 1.06141, 0.898203,- 1.34439, 0.302369] ,$$

$$n=4 ,$$

$$R_{{i}}= \left( \rho \left( a \right) \right) ^{t_{{i}}}+ \left( \rho \left( b \right) \right) ^{t_{{i}}} ,$$

$$f=\sum _{i=1}^{n}\omega_{{i}}R_{{i}} .$$

J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003).

J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev. Lett. 91, 146401 (2003).

$$p=[- 0.98, 0.3271, 0.7035] ,$$

$$q=[- 0.003557,- 0.03229, 0.007695] ,$$

$$r=[ 0.00625,- 0.02942, 0.05153] ,$$

$$t=[- 0.00002354, 0.002134, 0.00003394] ,$$

$$u=[- 0.0001283,- 0.005452,- 0.001269] ,$$

$$v=[ 0.0003575, 0.01578, 0.001296] ,$$

$$\alpha=[ 0.001867, 0.005151, 0.00305] ,$$

$$g= \left( \rho \left( s \right) \right) ^{4/3}F \left( \chi \left( s \right) ,{\it zs},p_{{1}},q_{{1}},r_{{1}},t_{{1}},u_{{1}},v_{{1}}, \alpha_{{1}} \right) +{\it ds}\,\epsilon \left( \rho \left( s \right) ,0 \right) F \left( \chi \left( s \right) ,{\it zs},p_{{2}},q_{{2}},r_{{2 }},t_{{2}},u_{{2}},v_{{2}},\alpha_{{2}} \right) ,$$

$$G= \left( \rho \left( s \right) \right) ^{4/3}F \left( \chi \left( s \right) ,{\it zs},p_{{1}},q_{{1}},r_{{1}},t_{{1}},u_{{1}},v_{{1}}, \alpha_{{1}} \right) +{\it ds}\,\epsilon \left( \rho \left( s \right) ,0 \right) F \left( \chi \left( s \right) ,{\it zs},p_{{2}},q_{{2}},r_{{2 }},t_{{2}},u_{{2}},v_{{2}},\alpha_{{2}} \right) ,$$

$$f=F \left( x,z,p_{{3}},q_{{3}},r_{{3}},t_{{3}},u_{{3}},v_{{3}},\alpha_{ {3}} \right) \left( \epsilon \left( \rho \left( a \right) ,\rho \left( b \right) \right) -\epsilon \left( \rho \left( a \right) ,0 \right) -\epsilon \left( \rho \left( b \right) ,0 \right) \right) ,$$

$$x= \left( \chi \left( a \right) \right) ^{2}+ \left( \chi \left( b \right) \right) ^{2} ,$$

$${\it zs}={\frac {\tau \left( s \right) }{ \left( \rho \left( s \right) \right) ^{5/3}}}-{\it cf} ,$$

$$z={\frac {\tau \left( a \right) }{ \left( \rho \left( a \right) \right) ^{5/3}}}+{\frac {\tau \left( b \right) }{ \left( \rho \left( b \right) \right) ^{5/3}}}-2\,{\it cf} ,$$

$${\it ds}=1-{\frac { \left( \chi \left( s \right) \right) ^{2}}{4\,{ \it zs}+4\,{\it cf}}} ,$$

$$F \left( x,z,p,q,c,d,e,f,\alpha \right) ={\frac {p}{\lambda \left( x,z, \alpha \right) }}+{\frac {q{x}^{2}+cz}{ \left( \lambda \left( x,z, \alpha \right) \right) ^{2}}}+{\frac {d{x}^{4}+e{x}^{2}z+f{z}^{2}}{ \left( \lambda \left( x,z,\alpha \right) \right) ^{3}}} ,$$

$$\lambda \left( x,z,\alpha \right) =1+\alpha\, \left( {x}^{2}+z \right) ,$$

$${\it cf}=3/5\,{3}^{2/3} \left( {\pi }^{2} \right) ^{2/3} ,$$

$$T=[ 0.031091, 0.015545, 0.016887] ,$$

$$U=[ 0.21370, 0.20548, 0.11125] ,$$

$$V=[ 7.5957, 14.1189, 10.357] ,$$

$$W=[ 3.5876, 6.1977, 3.6231] ,$$

$$X=[ 1.6382, 3.3662, 0.88026] ,$$

$$Y=[ 0.49294, 0.62517, 0.49671] ,$$

$$P=[1,1,1] ,$$

$$\epsilon \left( \alpha,\beta \right) = \left( \alpha+\beta \right) \left( e \left( l \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}} ,W_{{1}},X_{{1}},Y_{{1}},P_{{1}} \right) -{\frac {e \left( l \left( \alpha,\beta \right) ,T_{{3}},U_{{3}},V_{{3}},W_{{3}},X_{{3}},Y_{{3}},P _{{3}} \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( 1- \left( \zeta \left( \alpha,\beta \right) \right) ^ {4} \right) }{c}}+ \left( e \left( l \left( \alpha,\beta \right) ,T_{{2 }},U_{{2}},V_{{2}},W_{{2}},X_{{2}},Y_{{2}},P_{{2}} \right) -e \left( l \left( \alpha,\beta \right) ,T_{{1}},U_{{1}},V_{{1}},W_{{1}},X_{{1}},Y _{{1}},P_{{1}} \right) \right) \omega \left( \zeta \left( \alpha,\beta \right) \right) \left( \zeta \left( \alpha,\beta \right) \right) ^{ 4} \right) ,$$

$$l \left( \alpha,\beta \right) =1/4\,\sqrt [3]{3}{4}^{2/3}\sqrt [3]{{ \frac {1}{\pi \, \left( \alpha+\beta \right) }}} ,$$

$$\zeta \left( \alpha,\beta \right) ={\frac {\alpha-\beta}{\alpha+\beta}} ,$$

$$\omega \left( z \right) ={\frac { \left( 1+z \right) ^{4/3}+ \left( 1-z \right) ^{4/3}-2}{2\,\sqrt [3]{2}-2}} ,$$

$$e \left( r,t,u,v,w,x,y,p \right) =-2\,t \left( 1+ur \right) \ln \left( 1+1/2\,{\frac {1}{t \left( v\sqrt {r}+wr+x{r}^{3/2}+y{r}^{p+1} \right) }} \right) ,$$

$$c= 1.709921 .$$

Automatically generated von Weizsäcker kinetic energy. $$g={\frac {c\sigma \left( {\it ss} \right) }{\rho \left( s \right) }} ,$$

$$G={\frac {c\sigma \left( {\it ss} \right) }{\rho \left( s \right) }} ,$$

$$c=1/8 .$$

VWN 1980(III) functional $$x=1/4\,\sqrt [6]{3}{4}^{5/6}\sqrt [6]{{\frac {1}{\pi \,\rho}}} ,$$

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

$$f=\rho\,e ,$$

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

$$l=[- 0.409286,- 0.743294,- 0.228344] ,$$

$$m=[ 13.0720, 20.1231, 1.06835] ,$$

$$n=[ 42.7198, 101.578, 11.4813] ,$$

$$e=\Lambda+z \left( \lambda-\Lambda \right) ,$$

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

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

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

$$q \left( A,p,c,d \right) =A \left( \ln \left( {\frac {{x}^{2}}{X \left( x,c,d \right) }} \right) +2\,c\arctan \left( {\frac {Q \left( c ,d \right) }{2\,x+c}} \right) \left( Q \left( c,d \right) \right) ^{- 1}-cp \left( \ln \left( {\frac { \left( x-p \right) ^{2}}{X \left( x,c ,d \right) }} \right) +2\, \left( c+2\,p \right) \arctan \left( {\frac {Q \left( c,d \right) }{2\,x+c}} \right) \left( Q \left( c,d \right) \right) ^{-1} \right) \left( X \left( p,c,d \right) \right) ^{-1} \right) ,$$

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

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

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

VWN 1980(V) functional. The fitting parameters for $\Delta\varepsilon_{c}(r_{s},\zeta)_{V}$ appear in the caption of table 7 in the reference. $$x=1/4\,\sqrt [6]{3}{4}^{5/6}\sqrt [6]{{\frac {1}{\pi \,\rho}}} ,$$

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

$$f=\rho\,e ,$$

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

$$l=[- 0.10498,- 0.325,- 0.0047584] ,$$

$$m=[ 3.72744, 7.06042, 1.13107] ,$$

$$n=[ 12.9352, 18.0578, 13.0045] ,$$

$$e=\Lambda+\alpha\,y \left( 1+h{\zeta}^{4} \right) ,$$

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

$$h=4/9\,{\frac {\lambda-\Lambda}{ \left( \sqrt [3]{2}-1 \right) \alpha}} -1 ,$$

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

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

$$\alpha=q \left( k_{{3}},l_{{3}},m_{{3}},n_{{3}} \right) ,$$

$$q \left( A,p,c,d \right) =A \left( \ln \left( {\frac {{x}^{2}}{X \left( x,c,d \right) }} \right) +2\,c\arctan \left( {\frac {Q \left( c ,d \right) }{2\,x+c}} \right) \left( Q \left( c,d \right) \right) ^{- 1}-cp \left( \ln \left( {\frac { \left( x-p \right) ^{2}}{X \left( x,c ,d \right) }} \right) +2\, \left( c+2\,p \right) \arctan \left( {\frac {Q \left( c,d \right) }{2\,x+c}} \right) \left( Q \left( c,d \right) \right) ^{-1} \right) \left( X \left( p,c,d \right) \right) ^{-1} \right) ,$$

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

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

Here it means M05 exchange-correlation part which excludes HF exact exchange term. Y. Zhao, N. E. Schultz, and D. G. Truhlar, J. Chem. Phys. 123, 161103 (2005).

Here it means M05-2X exchange-correlation part which excludes HF exact exchange term. Y. Zhao, N. E. Schultz, and D. G. Truhlar, J. Chem. Theory Comput. 2, 364 (2006).

Here it means M06 exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, Theor. Chem. Acc. 120, 215 (2008).

Here it means M06-2X exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, Theor. Chem. Acc. 120, 215 (2008).

Here it means M06-HF exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, J. Phys. Chem. A 110, 13126 (2006).

Y. Zhao and D. G. Truhlar, J. Chem. Phys. 125, 194101 (2006).

Here it means M08-HX exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, J. Chem. Theory Comput. 4, 1849 (2008).

Here it means M08-SO exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, J. Chem. Theory Comput. 4, 1849 (2008).

R. Peverati and D. G. Truhlar, Journal of Physical Chemistry Letters 3, 117 (2012).

Y. Zhao and D. G. Truhlar, J. Chem. Phys. 128, 184109 (2008).

R. Peverati, Y. Zhao and D. G. Truhlar, J. Phys. Chem. Lett. 2 (16), 1991 (2011).

Here it means SOGGA11-X exchange-correlation part which excludes HF exact exchange term. R. Peverati and D. G. Truhlar, J. Chem. Phys. 135, 191102 (2011).