Aluminium KS-DFT versus Thomas–Fermi

This capability example compares the Kohn–Sham density-functional-theory (KS-DFT, internally qm) and finite-temperature Thomas–Fermi (TF) electronic backends in Otter. Both electronic models feed the same ion-sphere pseudoatom, QOZ/HNC, and Chabrier-1990 local-field-correction construction. The sequence

\[\rho=8.1\ {\rm g\,cm^{-3}},\qquad T_e=T_i\in\{1,15,50,100\}\ {\rm eV}\]

therefore isolates temperature-dependent electronic-model sensitivity without simultaneously changing the ion density.

The TF construction follows Eqs. (2)–(11) of Starrett and Saumon [2014], while the finite-temperature jellium local-field correction follows Chabrier [1990]. These citations define the methods; no numerical curve is extracted from either paper.

Example and execution modes

Aluminium KS-DFT and Thomas–Fermi comparison

Running the example calculates all eight QM/TF states from physical inputs:

poetry run python docs/examples/plot_al_qm_tf.py

The default is RECOMPUTE_WITH_OTTER = True. Results are written under benchmarks/outputs/al_qm_tf/gallery_recomputed; no bundled NPZ is needed.

What is plotted

The electronic figure compares:

  • \(4\pi r^2[n_{\rm full}(r)-n_0]\) for KS-DFT and TF, shown over \(-1\leq r\leq8\) Bohr; and

  • \(4\pi r^2n_{\rm scr}(r)\) for both models, shown over \(-1\leq r\leq12\) Bohr.

The negative lower limits are visual margins only; the radial grids begin at positive \(r\). The ionic-density partition is available in the NPZ export. Separate panels show the resulting \(g_{ii}(r)\) and \(S_{ii}(k)\).

Temperature sequence

At 1 and 15 eV, the orbital shell structure produces a large difference in the pseudoatom-partition ionization and in the screening cloud. The difference decreases at 50 eV. By 100 eV, the two ionic structures are very close for this state even though the electronic decompositions are not identical. The four points illustrate a trend; they do not define a universal temperature boundary for the validity of TF theory.

The table below uses the September 2026 reviewed v2 files. Delta Z is \(\bar Z_{\rm TF}-\bar Z_{\rm KS-DFT}\) using the QOZ pseudoatom-partition ionization. RMSEs use \(r\leq12\) Bohr and \(k\leq6\) Bohr-1.

\(T\) [eV]

\(\Delta Z\)

RMSE \(g_{ii}\)

RMSE \(S_{ii}\)

1

+1.84369

0.06051

0.07196

15

+1.81326

0.04190

0.03437

50

+0.66385

0.01167

0.01078

100

-0.19889

0.00280

0.00249

Numerical record

The four pickle-free NPZ files record converged full/external AA stages, threshold-state status, HNC residuals below \(10^{-6}\), transform-closure mismatches below \(10^{-3}\), and the model settings and checksums needed to reproduce the figures. Arrays are restricted to \(r,k<20\) in the following units:

  • density: Bohr-3

  • radius: Bohr

  • wavenumber: Bohr-1

  • chemical and real-space potential energy: Hartree

  • reciprocal-space pair potential: Hartree Bohr3

  • \(g_{ii}\) and \(S_{ii}\): dimensionless

The API model labels are ("qm", "tf"); figures identify qm as KS-DFT.