Atomic Reference Data for Electronic Structure Calculations

LaTeX Version of Equations

Eq. 2
$$r_s=\left({{3}\over{4\pi n}}\right)^{1/3} ~ ,$$

Eq. 3
$$\zeta=(n_\uparrow-n_\downarrow)/n ~ ,$$

Eq. 4
$$\varepsilon_x^P(r_s)= 2^{-1/3} \varepsilon_x^F(r_s) = -3 \left({{9}\over{32\pi^2}}\right)^{1/3} r_s^{-1} ~ ,$$

Eq. 5
$$f(\zeta) = {{ (1+\zeta)^{4/3} + (1-\zeta)^{4/3} - 2} \over {2(2^{1/3}-1)}} ~ ;$$

Eq. 6
\begin{eqnarray} F(r_s; A, x_0, b, c) & = & A \Big\{ \ln {{x^2}\over{X(x)}} + {{2b}\over{Q}} \tan^{-1} {{Q}\over{2x+b}} \nonumber \\ & & - {{b x_0}\over{X(x_0)}} \Big[ \ln {{(x-x_0)^2}\over{X(x)}} + {{2(b+2x_0)}\over{Q}} \tan^{-1} {{Q}\over{2x+b}} \Big] \Big\}, \nonumber \end{eqnarray}

Eq. 7
$$\varepsilon_c (r_s,\zeta) = \varepsilon_c^P (r_s) + \Delta \varepsilon_c (r_s,\zeta) ~ .$$

Eq. 8
$$\Delta \varepsilon_c (r_s,\zeta) = \alpha_c(r_s) [ f(\zeta)/f^{\prime\prime}(0)][1+\beta(r_s)\zeta^4] ~ ,$$

Eq. 9
$$\beta(r_s) = {{f''(0) \Delta \varepsilon_c(r_s,1)}\over{\alpha_c(r_s)}} -1\quad,$$

Eq. 10
$$\Delta \varepsilon_c(r_s,1) = \varepsilon_c(r_s,1) - \varepsilon_c(r_s,0) = \varepsilon_c^F(r_s) - \varepsilon_c^P(r_s)$$

Eq. 11
$$V_{xc}(n) = {{{\rm d} [ n (\varepsilon_x + \varepsilon_c)]}\over{{\rm d} n}} ~ .$$

Eq. 12
$$E_{\rm xc}[n] = E_{\rm x}^{\rm DF}[n] + E_{\rm x}^{\rm T}[n] + E_{\rm c}[n]$$

Eq. 13
$$\varepsilon_x^{\rm DF}(n) = \varepsilon(n) \phi_{\rm C }(n)$$

Eq. 14
$$\varepsilon_{\rm x}^{\rm T}(n) = \varepsilon(n) \phi_{\rm T} (n) .$$

Eq. 15
$$\phi_C (n) = \left[ {{5} \over{6}} + {{1} \over{3\beta^2}} + {{2\eta \ln \xi} \over{3\beta}} - {{2\eta^4 \ln \eta}\over{3\beta^4}} - {1\over2} \left( {{\beta \eta - \ln \xi}\over{\beta^2}} \right)^2 \right]$$

Eq. 16
$$\phi_T (n) = \left[ {1\over6} - {{1}\over{3\beta^2}} - {{2\eta \ln \xi}\over{3\beta}} + {{2\eta^4\ln \eta}\over{3\beta^4}} - \left( {{\beta \eta - \ln \xi}\over{\beta^2}} \right)^2 \right] ,$$

Eq. 17
$$\beta = v_F/c = (\hbar/(mc))(3\pi^2 n)^{1/3},$$

Eq. 18
$$\eta = (1+\beta^2)^{(1/2)} ;$$

Eq. 19
$$\xi = \beta + \eta .$$

Eq. 20
$$ \phi_{\rm C}(n) + \phi_{\rm T}(n) = \left[ 1 - {3\over2} \left( {{\beta \eta - \ln \xi}\over{\beta^2}} \right)^2 \right].$$

Eq. 21
$$E_{xc}[n] = \varepsilon(n) [\phi_C (n) +\phi_T (n)] + E_c[n].$$

Eq. 22
$$r_n = r_{\rm min} \left({{r_{\rm max}}\over{r_{\rm min}}}\right)^{n/N} $$

Eq. 23
$$r_n = a ({\rm e}^{b (n-1)} - 1) ~ ,$$

Eq. 24
$$\rho = \ln r. $$

Eq. 25
$$[-{1\over2}\nabla^2 + v_{\rm eff}(\vec r)] \psi_i(\vec r) = \varepsilon_i \psi_i(\vec r) ~ ,$$

Eq. 26
$$v_{\rm eff}(\vec r) = v(\vec r) + \int {\rm d}\vec r^\prime {{\rho(\vec r)}\over {|\vec r - \vec r' |}} + v_{\rm xc} (\vec r) ~ .$$

Eq. 27
$$\rho (\vec r) = 2 \sum_i f_i \mid \psi_i (\vec r)\mid^2 ~.$$

Eq. 28
$$ T = - 2 \sum_i f_i \int {\rm d}\vec{r} \, \psi_i^\ast (\vec{r}) \, {\textstyle{1\over2}} \, \nabla^2 \psi_i (\vec{r}) ~,$$

Eq. 29
$$ E_{\rm enuc} = \int {\rm d}{\vec{r}} \rho(\vec{r}) \, v(\vec{r}) ~ ,$$

Eq. 30
$$ E_{\rm coul} = \, {\textstyle{1\over2}} \, \int {\rm d}\vec{r} {\rm d}\vec{r}^\prime ~ {{\rho(\vec{r}) \rho(\vec{r}^\prime) }\over{\mid \vec{r} - \vec{r}^\prime\mid }} ~ ,$$

Eq. 31
$$E_{\rm xc} = \int d \vec r \rho(\vec r) \varepsilon_{\rm xc}(\rho) ~ ,$$

Eq. 32
$${{{\rm d}F}\over{{\rm d}r}} - {\kappa\over r} F= -c^{-1} (\epsilon-v(r) ) G , $$

Eq. 33
$${{{\rm d}G}\over{{\rm d}r}} + {\kappa\over r} G= c^{-1} (\epsilon-v(r)+2 c^2) F ,$$

Eq. 34
$$\psi = \pmatrix {G(r) r^{-1} {\cal Y}_{\kappa m} (\hat r) \cr
i F(r) r^{-1} {\cal Y}_{-\kappa m} (\hat r) \cr }$$

$\rho(\vec{r})= 2\sum_i\,f_i \sum_\mu |\psi_\mu(\vec{r})|^2$

Eq. 35
$${{{\rm d}^2 G}\over{{\rm d}r^2}} - {{\ell(\ell+1)}\over{r^2}}~ G = 2 M (V-\epsilon) G + {{1}\over{M}} {{{\rm d} M}\over{{\rm d}r}} \left( {{{\rm d}G}\over{{\rm d}r}} + {{\langle \kappa \rangle}\over{r}} \right) ~ ,$$

Eq. 36
$$M = 1 + {{\alpha^2}\over{2}} (\epsilon-V) ~ ,$$

Eq. 37
$$r^2 \rho(r) = G(r)^2$$