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Atomic Reference Data for Electronic Structure Calculations, The Exchange Term

The exchange term is given by

$$\varepsilon_x(r_s,\zeta) = \varepsilon_x^P (r_s) + [\varepsilon_x^F (r_s) - \varepsilon_x^P(r_s) ] f(\zeta) ~ .$$

(Eq. 1)

The electron gas parameter, rs, the spin polarization, ζ, and the ferromagnetic and paramagnetic exchange energies, $$\varepsilon_x^F (r_s), \varepsilon_x^P(r_s)$$

are defined as

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

(Eq. 2)

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

(Eq. 3)

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

(Eq. 4)

and f(ζ) is given by

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

(Eq. 5)

where n is the electron number density (implicitly a function of the spatial coordinates), and $$n_\uparrow$$ and $$n_\downarrow$$ its corresponding spin-up and spin-down components $$(n = n_\uparrow + n_\downarrow).$$

The correlation term

The correlation term is related to the function,

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

(Eq. 6)

where we have x = rs1/2,   X(x) = x2+bx+c,   Q = (4c-b2)1/2. The parameters x0, b, and c, given in the table below, are used to create three instances of F, using the table below.

    A x0 b c
Paramagnetic εcP 0.031 090 7 -0.104 98 3.727 44 12.935 2
Ferromagnetic εcF 0.015 545 35 -0.325 00 7.060 42 18.057 8
Spin Stiffness αc -1/(6π2) -0.004 758 40 1.131 07 13.004 5

The correlation energy is given by

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

(Eq. 7)

The paramagnetic correlation energy is given by εcP(rs) = F(rs; A, x0, b, c) with the four parameters taken from the "Paramagnetic" line in the table. (Similar definitions hold for εcF and αc.) The polarization contribution is given by

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

(Eq. 8)

with the function f as defined above, and where we have

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

(Eq. 9)


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

(Eq. 10)

is the difference of the ferromagnetic and paramagnetic correlation energies.

The exchange-correlation potential is given by

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

(Eq. 11)

We use this form in all of the codes in this study. To avoid undetected code bugs, the associated subroutine was recoded independently for one of the codes, although the other three codes shared a common subroutine. This subroutine will be provided upon request to

Created October 15, 2015, Updated November 4, 2019