Van-der-Waals exchange-correlation functionals and their high pressure and warm dense matter applications

We investigate basic hydrogen quantities like the molecular bond length, the molecular dissociation energy and the van-der-Waals interaction in idealized situations in an effort to discern a suitable exchange-correlation functional for the molecular to metal transition in warm dense hydrogen. The best reproduction of bond length and dissociation energy is given by the r2SCAN functional, several vdW functionals and also HSE06 fair qualitatively and quantitatively no better than PBE or worse. In addition we investigate quantities like the static and dynamic ion structure factor, and the electronic DOS to determine differences between exchange-correlation functionals with and without van-der-Waals corrections in the transition region from the molecular to the metallic regime of hydrogen.

💡 Research Summary

This paper presents a systematic benchmark of several exchange‑correlation (xc) functionals for dense hydrogen, with a focus on the molecular‑to‑metal transition that occurs under high‑pressure and warm‑dense‑matter (WDM) conditions. The authors first assess basic molecular properties—H₂ bond length, dissociation energy, and the van‑der‑Waals (vdW) interaction between two isolated H₂ molecules—using density‑functional theory (DFT) and comparing the results to highly accurate quantum Monte‑Carlo (i‑FCIQMC) reference data. For the bond length, the meta‑GGA r2SCAN reproduces the experimental value within 0.001 Å, outperforming both LDA and PBE, which overestimate the bond length by 0.10–0.25 Å. In terms of dissociation energy, r2SCAN again shows the smallest deviation (≈0.07 eV) whereas PBE underestimates by ≈0.21 eV and the three vdW‑DF variants overestimate by about 0.25 eV. The hybrid functional HSE06 yields results comparable to r2SCAN for these static quantities but at a substantially higher computational cost.

The second part of the study investigates the vdW interaction between two H₂ molecules in several relative orientations (geometries 1–5). The QMC reference curve exhibits a shallow minimum of roughly 5 meV at an intermolecular distance of ≈2.95 Å. PBE reproduces the position of this minimum well, though its depth is about 25 % larger than the QMC value. Among the vdW‑DF functionals, vdW‑DF2 behaves similarly to PBE, while vdW‑DF1 and vdW‑DF3 place the minimum at larger separations and produce overly deep wells. HSE06 matches the QMC curve almost perfectly across all geometries. Strikingly, r2SCAN fails to generate any minimum at all, indicating that despite its good performance for isolated‑molecule energetics it does not capture long‑range dispersion forces in hydrogen.

Having established the performance on isolated systems, the authors turn to realistic WDM conditions by generating equation‑of‑state (EOS) data over a wide range of densities and temperatures (including the molecular regime, the interacting molecular regime, and the pressure‑induced dissociation regime). At low densities the EOS curves of all functionals are essentially indistinguishable; differences are dominated by numerical uncertainties. At high densities, however, the vdW‑DF functionals systematically predict higher pressures (by 5–10 %) than PBE for the same temperature and density. This shift is consistent with the earlier observation that vdW‑DF functionals over‑stabilize the H₂ molecule, raising the dissociation energy and thus requiring more work to break the bond. Consequently, the liquid‑liquid phase transition (LLPT) line moves to higher pressures/temperatures when vdW corrections are included. The authors note, however, that the vdW contribution itself is only a few meV, whereas the total binding energies are on the order of several eV, so the overall impact on the thermodynamics is modest.

Static structure factors S(k), dynamic structure factors S(k, ω), and electronic density‑of‑states (DOS) are also examined across the transition. No significant functional‑dependent differences are observed; the electronic DOS shows the expected closure of the band gap at the molecular‑metal transition for all functionals, indicating that vdW corrections do not materially affect the electronic structure in this regime.

From a practical standpoint, the computational cost analysis shows that HSE06 is roughly 5–10 times more expensive than PBE, while vdW‑DF2 incurs only a modest overhead relative to PBE. Given that r2SCAN fails to describe dispersion and that HSE06’s advantage is limited to a more accurate vdW curve, the authors recommend PBE as the most efficient baseline functional for WDM hydrogen simulations. When a more accurate description of vdW forces is required, HSE06 is the preferred choice despite its higher cost; vdW‑DF2 offers a reasonable compromise between accuracy and efficiency.

In summary, the paper concludes that: (i) r2SCAN is optimal for reproducing isolated‑molecule bond lengths and dissociation energies but is unsuitable for dispersion‑driven phenomena; (ii) HSE06 provides the best overall agreement with QMC for vdW interactions, closely followed by PBE; (iii) vdW‑DF functionals tend to over‑stabilize molecules, shifting the EOS and LLPT to higher pressures; and (iv) for large‑scale WDM simulations of hydrogen, PBE offers the best balance of accuracy, computational cost, and robustness, with HSE06 reserved for studies where precise vdW energetics are essential.