* 交換増強因子の表式とパラメータ [#l39be222] 交換エネルギー汎関数 \[ E_{xc}[n]=\int{\rm d}\bm{r}\,n\varepsilon_{x}^{\rm LDA}(n)F_{x}(s),\quad E_{x}[n]=\int{\rm d}\bm{r}\,n\varepsilon_{x}^{\rm LDA}(n)F_{x}(s),\quad \varepsilon_{x}^{\rm LDA}(n)=-\frac{3}{4\pi}k_F,\quad k_F=(3\pi^2n)^{1/3},\quad s=\frac{|\nabla n|}{2k_Fn} \] における$F_x(s)$を交換増強因子と呼ぶ。 ** B86b型 [#zfe74a91] \[ F_x(s)=1+\frac{\mu s^2}{(1+\mu s^2/\kappa)^{4/5}}=\left\{ \begin{array}{ll} 1+\mu s^2&s\rightarrow0\\ (\mu\kappa^4)^{1/5}s^{2/5}&s\rightarrow\infty \end{array} \right. \] |CENTER:Method|CENTER:$\mu$|CENTER:$\kappa$|CENTER:$(\mu\kappa^4)^{1/5}$|CENTER:Reference|h |B86b|0.2449|0.5757|0.485|[[A. D. Becke, JCP 85, 7184 (1986):http://scitation.aip.org/content/aip/journal/jcp/85/12/10.1063/1.451353]]| |optB86b|0.1234|1|0.658|[[J. Klimes et al., PRB 83, 195131 (2011):http://journals.aps.org/prb/abstract/10.1103/PhysRevB.83.195131]]| |B86R|0.1234|0.7114|0.501|[[I. Hamada, PRB 89, 121103(R) (2014):http://journals.aps.org/prb/abstract/10.1103/PhysRevB.89.121103]]| ただし$\mu_{\rm GEA}=10/81\simeq0.1234$である。 ** PW86型 [#jb0f34cb] \[ F_x(s)=(1+15as^2+bs^4+cs^6)^{1/15}=\left\{ \begin{array}{ll} 1+a s^2&s\rightarrow0\\ c^{1/15}s^{2/5}&s\rightarrow\infty \end{array} \right. \] |CENTER:Method|CENTER:$a$|CENTER:$b$|CENTER:$c$|CENTER:$c^{1/15}$|CENTER:Reference|h |PW86|0.0864|14|0.2|0.898|[[J. P. Perdew and Y. Wang, PRB 33, 8800(R) (1986):http://journals.aps.org/prb/abstract/10.1103/PhysRevB.33.8800]]| |PW86R|0.1234|17.33|0.163|0.886|[[E. D. Murray et al., JCTC 10, 2754 (2009):http://pubs.acs.org/doi/abs/10.1021/ct900365q]]| ただし$7/81\simeq0.0864, \mu_{\rm GEA}=10/81\simeq0.1234$である。 ** B88型 [#z125b612] \[ F_x(s)=1+\frac{\mu s^2}{1+\frac{9}{4\pi}\mu s\,{\rm sinh}^{-1}(\lambda s)} \] |CENTER:Method|CENTER:$\mu$|CENTER:$\lambda$|CENTER:Reference|h |B88|$0.2743$|$2(6\pi^2)^{1/3}$|[[A. D. Becke, Phys. Rev. A 38, 3098 (1988):http://journals.aps.org/pra/abstract/10.1103/PhysRevA.38.3098]]| |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 [#udd45bd2] ** PBE型 [#a62f0db3] \[ F_x(s)=1+\kappa-\frac{\kappa}{1+\mu s^2/\kappa}=1+\frac{\mu s^2}{1+\mu s^2/\kappa}=\left\{ \begin{array}{ll} 1+\mu s^2&s\rightarrow0\\ 1+\kappa&s\rightarrow\infty \end{array} \right. \] |CENTER:Method|CENTER:$\mu$|CENTER:$\kappa$|CENTER:Reference|h ||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||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)| ただし$\mu_{\rm GEA}=10/81\simeq0.1235$である。 ** RPBE型 [#v8c3ded0] \[ F_x(s)=1+\kappa(1-e^{-\mu s^2/\kappa})=\left\{ \begin{array}{ll} 1+\mu s^2&s\rightarrow0\\ 1+\kappa&s\rightarrow\infty \end{array} \right. \] |CENTER:Method|CENTER:$\mu$|CENTER:$\kappa$|CENTER:Reference|h |RPBE|0.21951|0.804|B. Hammer et al. Phys. Rev. B 59, 7413 (1999)| ** C09型 [#o170764e] \[ F_x(s)=1+\mu s^2 e^{-\alpha s^2}+\kappa(1-e^{-\alpha s^2/2})=\left\{ \begin{array}{ll} 1+(\mu+\kappa\alpha/2) s^2&s\rightarrow0\\ 1+\kappa&s\rightarrow\infty \end{array} \right. \] |CENTER:Method|CENTER:$\mu$|CENTER:$\kappa$|CENTER:$\alpha$|CENTER:Reference|h |C09|0.0617|1.245|0.0483|V. R. Cooper Phys. Rev. B 81, 161104(R) (2010)|