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Assumptions:

  1. interacting system

  2. uniform gas system

In the Kohn-Sham model, the exchange correlation functional is introduced to account for correlation energy $\epsilon_{xc} = \epsilon_{x} + \epsilon_{c}$ is the exchange correlation energy per particle for a uniform electron gas where $\epsilon_{x}$ is the exchange energy per particle for a uniform electron gas and $\epsilon_{c}$ is the correlation energy per particle for a uniform electron gas.

The correlation energy of an interacting electron gas system has analytic solutions only in the limiting cases:

$\lim_{\rho \rightarrow 0}: \epsilon_{c} (\rho) = 0.311ln(r_{s}) - 0.048 + r_{s}(A^{0}ln(r_{s})++C^{0}) \rightarrow r_{s} >> 1$

$\lim_{\rho \rightarrow \infty}: \epsilon_{c}(\rho)=0.5[\frac{g_{0}}{r_{s}} + \frac{g_{1}}{r_{s}^{3/2}} + ...]$

However, for regions within the bounds of the limiting case, the Vosco-Wilk-Nusair correlation functional is invoked.

$\epsilon_{c}(r_{s})=\frac{A}{2}[ln(\frac{x}{X(x)} + \frac{2b}{Q}tan^{-1}(\frac{Q}{2x+b})-\frac{bx_{0}}{X(x_{0})}[ln(\frac{(x-x_{0})^{2}}{X(x)})+\frac{2(b+2x_{0})}{Q}tan^{-1}(\frac{Q}{2x+b})]]$

What does $A, r_{s}, x, X(x), Q$ refers to?

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1 Answer 1

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A very quick google shows up https://math.nist.gov/DFTdata/atomdata/node5.html#SECTION00021200000000000000 amongst other hits - note the definition is slightly different from yours and I think you want $x^2$ in the first term above, see also https://www.molpro.net/manual/doku.php?id=density_functional_descriptions. Quoting the first, with images where appropriate as it will be non-trivial to reformat:

The correlation term is related to the function, enter image
description here

where we have $x={r_s}^{1 \over 2}$, $X( x )=x^2+bx+c$, $Q=(4c-b^2)^{1 \over 2}$. The parameters $x_0$, b and c, given in the table below, are used to create three instances of F, using the table below.

enter image description here

$r_s$ is described on https://math.nist.gov/DFTdata/atomdata/node4.html#SECTION00021100000000000000. Again quoting with images hwere appropriate

The electron gas parameter $r_s$ ... are defined as

enter image description here

... where n is the electron number density (implicitly a function of the spatial coordinates)

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