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Disclaimer: Assume the person asking is a novice in DFT and condensed matter physics in general.

In the localized density approximation (LDA), the exchange correlation potential is considered a function of a constant charge density, whereas in GGA, it is a function of the gradient of charge density as well:

\begin{equation} V_{(xc)}^{LDA} = f(\rho_{(r)}) \end{equation}

\begin{equation} V_{(xc)}^{GGA} = f(\rho_{(r)},\nabla\rho_{(r)}) \end{equation}

What role does the constant charge density play in GGA? Also, in literature, methods describe DFT calculations done with GGA as parameterized by so-and-so. What does that mean?

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What role does the constant charge density play in GGA?

Most GGAs give out LDAs at the limit of zero gradient. This is, however, not always true.

Also, in literature, methods describe DFT calculations done with GGA as parameterized by so-and-so. What does that mean?

GGAs is a form of functional, like LDA: it does not say anything about which functional form is used. You can fit i.e. parametrize the functional in an infinite number of ways. This is an issue already at the LDA level: even though the exchange functional can be derived easily, the correlation part is only known at certain points derived from e.g. quantum Monte Carlo simulations. Some sort of interpolation is necessary to get a functional that can be evaluated at arbitrary densities, e.g. LibXC has almost 40 different LDA correlation functionals for three dimensional systems. If you go to the GGA level, the exchange functionals also have different forms.

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