Carbon local-field-correction sensitivity

This example isolates the static electron local-field correction (LFC) for carbon at \(\rho=5\ {\rm g\,cm^{-3}}\) and \(T_e=T_i=2,100\) eV. At each temperature all model branches reuse the same converged KS average atom, \(\bar Z\), screening cloud \(q(k)=n_{\rm scr}(k)\), and finite-temperature Lindhard response \(\chi^0_{ee}(k)\). Only the LFC and downstream effective potential and HNC calculation change.

The compared models are RPA (\(G_{ee}=0\)), Hubbard [Hubbard, 1958], Utsumi–Ichimaru [Utsumi and Ichimaru, 1982], Chabrier-1990 [Chabrier, 1990], and the Gregori finite-temperature interpolation [Geldart and Vosko, 1966, Gregori et al., 2007]. Chabrier-1990 is the comparison reference, not an asserted exact result. The pseudoatom/QOZ construction follows Starrett and Saumon [2014].

Why similar responses can give different potentials

The Starrett–Saumon reduction contains the inverse response [Starrett and Saumon, 2014],

\[V_{ii}(k)=\frac{4\pi\bar Z^2}{k^2} +\frac{q(k)^2}{\chi_{ee}(k)}.\]

For the LFC models used here, \(G_{ee}(k)=a k^2+O(k^4)\), so their interacting responses have the same leading Coulomb limit,

\[\chi_{ee}(k)=-\frac{k^2}{4\pi}+O(k^4).\]

The LFC contribution to the effective potential nevertheless approaches the finite, model-dependent limit

\[V_{\rm LFC}(k) =\frac{4\pi}{k^2}G_{ee}(k)q(k)^2 \longrightarrow 4\pi a\bar Z^2.\]

This is why response curves that nearly overlap at small \(k\) can produce visibly different \(V_{ii}(k)\). The example plots the stable decomposition

\[V_{ii}=V_{\rm charge}+V_{\rm LFC}+V_{\chi_0}\]

where, with \(d(k)=q(k)-\bar Z\),

\[\begin{split}V_{\rm charge} &= \frac{4\pi}{k^2} \left[-2\bar Z d(k)-d(k)^2\right],\\ V_{\rm LFC} &= \frac{4\pi}{k^2}G_{ee}(k)q(k)^2,\\ V_{\chi_0} &= \frac{q(k)^2}{\chi^0_{ee}(k)}.\end{split}\]

All branches share \(V_{\rm charge}+V_{\chi_0}\) because the average atom, \(q(k)\), and \(\chi^0_{ee}(k)\) are held fixed. This sum is used only to verify the decomposition and is not plotted as a model curve. Only \(V_{\rm LFC}\) changes; the complete three-term sum is passed to QOZ/HNC.

Run the example

Carbon: local-field-correction sensitivity

The source contains one user-facing switch:

RECOMPUTE_WITH_OTTER = False

False verifies and loads checksummed Otter results. True runs the shared electronic calculation and all five LFC/QOZ/HNC branches, then writes new files under benchmarks/outputs/carbon_lfc_sensitivity/gallery_recomputed. Existing accepted results are not overwritten.

The pickle-free NPZ files retain arrays through \(r,k\leq20\) in Bohr units. The manifest under benchmarks/baselines/carbon_lfc_sensitivity records the resolved settings and checksums.