交換増強因子の表式とパラメータ

B86b型

\[ F_x(s)=1+\frac{\mu s^2}{(1+\mu s^2/\kappa)^{4/5}}=\left\{ \begin{array}{ll} 1+\mu s^2&s\rightarrow0\\ (\mu\kappa^4)^{1/5}s^{2/5}&s\rightarrow\infty \end{array} \right. \]
Method\( \mu \)\( \kappa \)\( (\mu\kappa^4)^{1/5} \)Reference
B86b\( 0.2449 \)\( 0.5757 \)\( 0.485 \)A. D. Becke, J. Chem. Phys. 85, 7184 (1986)
optB86b\( 0.1234 \) \( (=10/81) \)\( 1 \)\( 0.658 \)J. Klimes et al., Phys. Rev. B 83, 195131 (2011)
B86R\( 0.1234 \) \( (=10/81) \)\( 0.7114 \)\( 0.501 \)I. Hamada, Phys. Rev. B 89, 121103(R) (2014)

PW86型

\[ F_x(s)=(1+15as^2+bs^4+cs^6)^{1/15}=\left\{ \begin{array}{ll} 1+a s^2&s\rightarrow0\\ c^{1/15}s^{2/5}&s\rightarrow\infty \end{array} \right. \]
Method\( a \)\( b \)\( c \)\( c^{1/15} \)Reference
PW86\( 0.0864 \)\( 14 \)\( 0.2 \)\( 0.898 \)J. P. Perdew and Y. Wang, Phys. Rev. B 33, 8800(R) (1986)
PW86R\( 0.1234 \) \( (=10/81) \)\( 17.33 \)\( 0.163 \)\( 0.886 \)E. D. Murray et al., J. Chem. Theory Comput. 10, 2754 (2009)

B88型

\[ F_x(s)=1+\frac{\mu s^2}{1+\frac{9}{4\pi}\mu s\,{\rm sinh}^{-1}(\lambda s)} \]
Method\( \mu \)\( \lambda \)Reference
B88\( 0.2743 \)\( 2(6\pi^2)^{1/3} \)A. D. Becke, Phys. Rev. A 38, 3098 (1988)
optB88\( 0.22 \)\( 1.2\times(9/2)(6/\pi)^{1/3} \)J. Klimes et al., J. Phys. Cond. Mat. 22, 022201 (2010)

PBE型

\[ F_x(s)=1+\frac{\mu s^2}{1+\mu s^2/\kappa}=\left\{ \begin{array}{ll} 1+\mu s^2&s\rightarrow0\\ 1+\kappa&s\rightarrow\infty \end{array} \right. \]
Method\( \mu \)\( \kappa \)Reference
\( 0.235 \)\( 0.967 \)A. D. Becke, J. Chem. Phys. 84, 4524 (1986)
PBE\( 0.21951 \)\( 0.804 \)J. P. Perdew et al., Phys. Rev. Lett. 77, 3865 (1996)
revPBE\( 0.21951 \)\( 1.245 \)Y. Zhang and W. Yang, Phys. Rev. Lett. 80, 890 (1998)
PBEsol\( 0.1235 \) \( (=10/81) \)J. P. Perdew et al., Phys. Rev. Lett. 100, 136406 (2008)
optPBE\( 0.175519 \)\( 1.04804 \)J. Klimes et al., J. Phys. Cond. Mat. 22, 022201 (2010)

RPBE型

\[ F_x(s)=1+\kappa(1-e^{-\mu s^2/\kappa}) \]
Method\( \mu \)\( \kappa \)Reference
RPBE\( 0.21951 \)\( 0.804 \)B. Hammer et al. 59, 7413 (1999)
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