Exchange term: Atomic Ref. Data Elec. Struct. Calculations

Atomic Reference Data
for Electronic Structure Calculations

The exchange term

The exchange term is given by

The electron gas parameter, r_s, the spin polarization, zeta , and the ferromagnetic and paramagnetic exchange energies, $\varepsilon_x^F(r_s),~ \varepsilon_x^P(r_s)$ are defined as

(Eq. 2)

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

(Eq. 3)

(Eq. 4)

and f( zeta ) is given by

(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,

(Eq. 6)

where we have x = r_s^1/2, 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	_c^P	0.031 090 7	-0.104 98	3.727 44	12.935 2
Ferromagnetic	_c^F	0.015 545 35	-0.325 00	7.060 42	18.057 8
Spin Stiffness	_c	-1/(6²)	-0.004 758 40	1.131 07	13.004 5

The correlation energy is given by

(Eq. 7)

The paramagnetic correlation energy is given by epsilon _c^P(r_s) = F(r_s; A, x₀, b, c) with the four parameters taken from the "Paramagnetic" line in the table. (Similar definitions hold for epsilon _c^F and alpha _c.) The polarization contribution is given by

(Eq. 8)

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

(Eq. 9)

and

(Eq. 10)

is the difference of the ferromagnetic and paramagnetic correlation energies.

The exchange-correlation potential is given by

(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 charles.clark@nist.gov.

Atomic Reference Data for Electronic Structure Calculations

The exchange term

The correlation term

Atomic Reference Data
for Electronic Structure Calculations