====== Density functional descriptions ======

===== B86: Xalpha beta gamma =====

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

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

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

$$\beta= 0.0076
,$$

$$\lambda= 0.004
.$$

===== B86MGC: Xalpha beta gamma with Modified Gradient Correction =====

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

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

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

$$\beta= 0.00375
,$$

$$\lambda= 0.007
.$$

===== B86R: Xalpha beta gamma Re-optimised =====

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

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

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

$$\beta= 0.00787
,$$

$$\lambda= 0.004
.$$

===== B88: Becke 1988 Exchange Functional =====

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

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

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

$$\beta= 0.0042
.$$

===== B88C: Becke 1988 Correlation Functional =====

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. $$f=- 0.8\,\rho \left( a \right) \rho \left( b \right) {q}^{2} \left( 1-{
\frac {\ln  \left( 1+q \right) }{q}} \right)
,$$

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

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

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

$$t= 0.63
,$$

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

$$z=2\,ur
,$$

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

$$u= 0.96
,$$

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

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

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

$$\beta= 0.00375
,$$

$$\lambda= 0.007
.$$

===== B95: Becke 1995 Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

$$l= 0.0031
,$$

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

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

$$\nu= 0.038
,$$

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

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

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

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

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

$$c= 1.709921
.$$

===== B97DF: Density functional part of B97 =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== B97RDF: Density functional part of B97 Re-parameterized by Hamprecht et al =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== BR: Becke-Roussel Exchange Functional =====

A. D. Becke and M. R. Roussel,[[https://dx.doi.org/10.1103/PhysRevA.39.3761|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 $$\gamma=1.$$

===== BRUEG: Becke-Roussel Exchange Functional — Uniform Electron Gas Limit =====

A. D. Becke and M. R. Roussel,[[https://dx.doi.org/10.1103/PhysRevA.39.3761|Phys. Rev. A]] **39**, 3761 (1989)

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

===== BW: Becke-Wigner Exchange-Correlation Functional =====

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

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

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

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

$$\beta= 0.0042
,$$

$$c= 0.04918
,$$

$$d= 0.349
.$$

===== CS1: Colle-Salvetti correlation functional =====

R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1974); C. Lee, W. Yang and R. G. Parr, [[https://dx.doi.org/10.1103/PhysRevB.37.785|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.

===== CS2: Colle-Salvetti correlation functional =====

R. Colle and O. Salvetti, Theor. Chim. Acta 37, 329 (1974); C. Lee, W. Yang and R. G. Parr, [[https://dx.doi.org/10.1103/PhysRevB.37.785|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 $a=0.04918, b=0.132, c=0.2533, d=0.349$.

===== DIRAC: Slater-Dirac Exchange Energy =====

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

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

===== ECERF: Short-range LDA correlation functional =====

Local-density approximation of correlation energy\\
for short-range interelectronic interaction ${\rm erf}(\mu r_{21})/r_{12}$,\\
S. Paziani, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, [[https://dx.doi.org/10.1103/PhysRevB.73.155111|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, [[https://dx.doi.org/10.1103/PhysRevB.45.13244|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, [[https://dx.doi.org/10.1103/PhysRevB.64.155102|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 $g_c(0,r_s,\zeta\!=\!0)=g(0,r_s,\zeta\!=\!0)-\frac{1}{2}$.

===== ECERFPBE: Range-Separated Correlation Functional =====

Toulouse-Colonna-Savin range-separated correlation functional based on PBE, see J. Toulouse et al., [[https://dx.doi.org/10.1063/1.1824896|J. Chem. Phys.]] **122**, 014110 (2005).

===== EXACT: Exact Exchange Functional =====

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

===== EXERF: Short-range LDA 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 $\alpha=(4/9\pi)^{1/3}$.

===== EXERFPBE: Range-Separated Exchange Functional =====

Toulouse-Colonna-Savin range-separated exchange functional based on PBE, see J. Toulouse et al., [[https://dx.doi.org/10.1063/1.1824896|J. Chem. Phys.]] **122**, 014110 (2005).

===== G96: Gill’s 1996 Gradient Corrected Exchange Functional =====

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

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

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

===== HCTH120: Handy least squares fitted functional =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== HCTH147: Handy least squares fitted functional =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== HCTH93: Handy least squares fitted functional =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== HJSWPBEX: Meta GGA Correlation Functional =====

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., [[https://dx.doi.org/10.1063/1.2921797|J. Chem. Phys.]] **128**, 194105 (2008).

===== LTA: Local tau Approximation =====

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

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

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

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

===== LYP: Lee, Yang and Parr Correlation Functional =====

C. Lee, W. Yang and R. G. Parr, [[https://dx.doi.org/10.1103/PhysRevB.37.785|Phys. Rev. B]] **37**, 785(1988); B. Miehlich, A. Savin, H. Stoll and H. Preuss, [[https://dx.doi.org/10.1016/0009-2614(89)87234-3|Chem. Phys. Lett.]] **157**, 200 (1989). $$f=-4\,A\rho \left( a \right) \rho \left( b \right)  \left( 1+{\frac {d}
{\sqrt [3]{\rho}}} \right) ^{-1}{\rho}^{-1}-AB\omega\, \left( \rho
 \left( a \right) \rho \left( b \right)  \left( 8\,{2}^{2/3}{\it cf}\,
 \left(  \left( \rho \left( a \right)  \right) ^{8/3}+ \left( \rho
 \left( b \right)  \right) ^{8/3} \right) + \left( {\frac {47}{18}}-{
\frac {7}{18}}\,\delta \right) \sigma- \left( 5/2-1/18\,\delta \right)
 \left( \sigma \left( {\it aa} \right) +\sigma \left( {\it bb} \right)
 \right) -1/9\, \left( \delta-11 \right)  \left( {\frac {\rho \left( a
 \right) \sigma \left( {\it aa} \right) }{\rho}}+{\frac {\rho \left( b
 \right) \sigma \left( {\it bb} \right) }{\rho}} \right)  \right) -2/3
\,{\rho}^{2}\sigma+ \left( 2/3\,{\rho}^{2}- \left( \rho \left( a
 \right)  \right) ^{2} \right) \sigma \left( {\it bb} \right) + \left(
2/3\,{\rho}^{2}- \left( \rho \left( b \right)  \right) ^{2} \right)
\sigma \left( {\it aa} \right)  \right)
,$$

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

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

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

$$A= 0.04918
,$$

$$B= 0.132
,$$

$$c= 0.2533
,$$

$$d= 0.349
.$$

===== M052XC: M05-2X Meta-GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

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

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

$$n=4
,$$

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

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

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

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

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

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

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

===== M052XX: M05-2X Meta-GGA Exchange Functional =====

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
,$$

$$n=11
,$$

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

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

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

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

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

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

===== M05C: M05 Meta-GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

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

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

$$n=4
,$$

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

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

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

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

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

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

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

===== M05X: M05 Meta-GGA Exchange Functional =====

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
,$$

$$n=11
,$$

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

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

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

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

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

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

===== M062XC: M06-2X Meta-GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

$$n=4
,$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===== M062XX: M06-2X Meta-GGA Exchange Functional =====

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
,$$

$$n=11
,$$

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

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

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

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

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

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

===== M06C: M06 Meta-GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

$$n=4
,$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===== M06HFC: M06-HF Meta-GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

$$n=4
,$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===== M06HFX: M06-HF Meta-GGA Exchange Functional =====

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
,$$

$$n=11
,$$

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

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

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

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

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

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

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

$$\alpha= 0.001867
,$$

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

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

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

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

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

===== M06LC: M06-L Meta-GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

$$n=4
,$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===== M06LX: M06-L Meta-GGA Exchange Functional =====

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
,$$

$$n=11
,$$

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

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

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

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

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

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

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

$$\alpha= 0.001867
,$$

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

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

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

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

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

===== M06X: M06 Meta-GGA Exchange Functional =====

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
,$$

$$n=11
,$$

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

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

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

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

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

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

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

$$\alpha= 0.001867
,$$

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

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

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

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

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

===== M12C: Meta GGA Correlation Functional =====

Meta-GGA correlation functional based on first principles, see M. Modrzejewski et al., [[https://dx.doi.org/10.1063/1.4768228|J. Chem. Phys.]] **137**, 204121 (2012).

===== MK00: Exchange Functional for Accurate Virtual Orbital Energies =====

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

===== MK00B: Exchange Functional for Accurate Virtual Orbital Energies =====

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

$$\beta= 0.0016
,$$

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

===== P86: . =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$z= 0.11
,$$

$$\alpha= 0.023266
,$$

$$\beta= 0.000007389
,$$

$$\xi= 8.723
,$$

$$\delta= 0.472
.$$

===== PBEC: PBE Correlation Functional =====

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

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

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

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

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

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

$$\iota= 0.0716
,$$

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

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

$$\kappa= 0.004235
,$$

$$Z=- 0.001667
,$$

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

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

$$\Xi= 23.266
,$$

$$\Phi= 0.007389
,$$

$$\Lambda= 8.723
,$$

$$\Upsilon= 0.472
,$$

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

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

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

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

===== PBESOLC: PBEsol Correlation Functional =====

===== PBESOLX: PBEsol Exchange Functional =====

===== PBEX: PBE Exchange Functional =====

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

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

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

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

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

$$R= 0.804
,$$

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

$$\delta= 0.066725
.$$

===== PBEXREV: Revised PBE Exchange Functional =====

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

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

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

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

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

$$R= 1.245
,$$

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

$$\delta= 0.066725
.$$

===== PW86: . =====

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

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

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

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

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

===== PW91C: Perdew-Wang 1991 GGA Correlation Functional =====

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

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

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

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

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

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

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

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

$$\iota= 0.09
,$$

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

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

$$\kappa= 0.004235
,$$

$$Z=- 0.001667
,$$

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

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

$$\Xi= 23.266
,$$

$$\Phi= 0.007389
,$$

$$\Lambda= 8.723
,$$

$$\Upsilon= 0.472
,$$

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
,$$

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

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

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

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

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

===== PW91X: Perdew-Wang 1991 GGA Exchange Functional =====

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

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

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

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

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

===== PW92C: Perdew-Wang 1992 GGA Correlation Functional =====

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== STEST: Test for number of electrons =====

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

===== TFKE: Thomas-Fermi Kinetic Energy =====

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

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

===== TH1: Tozer and Handy 1998 =====

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

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

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

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

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

$$n=21
,$$

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

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

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

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

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

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

===== TH2: . =====

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

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

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

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

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

$$n=19
,$$

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

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

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

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

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

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

===== TH3: . =====

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

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

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

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

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

$$n=19
,$$

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

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

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

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

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

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

===== TH4: . =====

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

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

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

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

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

$$n=19
,$$

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

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

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

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

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

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

===== THGFC: . =====

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 $D=0.1863$. $$t=[7/6,4/3,3/2,5/3,4/3,3/2,5/3,{\frac {11}{6}},3/2,5/3,{\frac {11}{6}},
2]
,$$

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

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

$$n=12
,$$

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

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

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

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

===== THGFCFO: . =====

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

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

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

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

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

$$n=20
,$$

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

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

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

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

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

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

===== THGFCO: . =====

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

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

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

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

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

$$n=20
,$$

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

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

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

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

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

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

===== THGFL: . =====

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

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

$$n=4
,$$

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

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

===== TPSSC: TPSS Correlation Functional =====

J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, [[https://dx.doi.org/10.1103/PhysRevLett.91.146401|Phys. Rev. Lett.]] **91**, 146401 (2003).

===== TPSSX: TPSS Exchange Functional =====

J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, [[https://dx.doi.org/10.1103/PhysRevLett.91.146401|Phys. Rev. Lett.]] **91**, 146401 (2003).

===== VSXC: . =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$c= 1.709921
.$$

===== VW: von Weizsäcker kinetic energy =====

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

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

$$c=1/8
.$$

===== VWN3: Vosko-Wilk-Nusair (1980) III local correlation energy =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===== VWN5: Vosko-Wilk-Nusair (1980) V local correlation energy =====

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

===== XC-M05: M05 Meta-GGA Exchange-Correlation Functional =====

Here it means M05 exchange-correlation part which excludes HF exact exchange term. Y. Zhao, N. E. Schultz, and D. G. Truhlar, [[https://dx.doi.org/10.1063/1.2126975|J. Chem. Phys.]] **123**, 161103 (2005).

===== XC-M05-2X: M05-2X Meta-GGA Exchange-Correlation Functional =====

Here it means M05-2X exchange-correlation part which excludes HF exact exchange term. Y. Zhao, N. E. Schultz, and D. G. Truhlar, [[https://dx.doi.org/10.1021/ct0502763|J. Chem. Theory Comput.]] **2**, 364 (2006).

===== XC-M06: M06 Meta-GGA Exchange-Correlation Functional =====

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).

===== XC-M06-2X: M06-2X Meta-GGA Exchange-Correlation Functional =====

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).

===== XC-M06-HF: M06-HF Meta-GGA Exchange-Correlation Functional =====

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).

===== XC-M06-L: M06-L Meta-GGA Exchange-Correlation Functional =====

Y. Zhao and D. G. Truhlar, [[https://dx.doi.org/10.1063/1.2370993|J. Chem. Phys.]] **125**, 194101 (2006).

===== XC-M08-HX: M08-HX Meta-GGA Exchange-Correlation Functional =====

Here it means M08-HX exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, [[https://dx.doi.org/10.1021/ct800246v|J. Chem. Theory Comput.]] **4**, 1849 (2008).

===== XC-M08-SO: M08-SO Meta-GGA Exchange-Correlation Functional =====

Here it means M08-SO exchange-correlation part which excludes HF exact exchange term. Y. Zhao and D. G. Truhlar, [[https://dx.doi.org/10.1021/ct800246v|J. Chem. Theory Comput.]] **4**, 1849 (2008).

===== XC-M11-L: M11-L Exchange-Correlation Functional =====

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

===== XC-SOGGA: SOGGA Exchange-Correlation Functional =====

Y. Zhao and D. G. Truhlar, [[https://dx.doi.org/10.1063/1.2912068|J. Chem. Phys.]] **128**, 184109 (2008).

===== XC-SOGGA11: SOGGA11 Exchange-Correlation Functional =====

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

===== XC-SOGGA11-X: SOGGA11-X Exchange-Correlation Functional =====

Here it means SOGGA11-X exchange-correlation part which excludes HF exact exchange term. R. Peverati and D. G. Truhlar, [[https://dx.doi.org/10.1063/1.3663871|J. Chem. Phys.]] **135**, 191102 (2011).