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第一原理分子動力学プログラム STATE Senri Wiki
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 | \( 0.0864 \) \( (=7/81) \) | \( 14 \) | \( 0.2 \) | \( 0.898 \) | 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) |
PW91型 †
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 †
\[ F_x(s)=1+\kappa(1-e^{-\mu s^2/\kappa})=\left\{ \begin{array}{ll} 1+\mu s^2&s\rightarrow0\\ 1+\kappa&s\rightarrow\infty \end{array} \right. \]
| Method | \( \mu \) | \( \kappa \) | Reference |
| RPBE | \( 0.21951 \) | \( 0.804 \) | B. Hammer et al. Phys. Rev. B 59, 7413 (1999) |
C09型 †
\[ F_x(s)=1+\mu s^2 e^{-\alpha s^2}+\kappa(1-e^{-\alpha s^2/2})=\left\{ \begin{array}{ll} 1+(\mu+\kappa\alpha/2) s^2&s\rightarrow0\\ 1+\kappa&s\rightarrow\infty \end{array} \right. \]
| Method | \( \mu \) | \( \kappa \) | \( \alpha \) | Reference |
| C09 | \( 0.0617 \) | \( 1.245 \) | \( 0.0483 \) | V. R. Cooper Phys. Rev. B 81, 161104(R) (2010) |
