We calculated T_c of metallic hydrogen considering the full momentum and energy dependence of the effective pairing interaction. The model for the interaction [1] assumes a Fermi sphere. Coulomb-screening, phonon dispersion and electron-phonon coupling are controlled by the density which, together with the nuclear mass, is the only parameter of the system. We solved the frequency and momentum dependent Eliashberg equations numerically -a surprisingly heavy computation- and obtained a maximum T_c of 0.13K at 10²¹cm^-3 [2].
A common approximation, consisting of replacing the k-dependence of the interaction with its value at the Fermi surface, gives much lower T_c’s with a maximum of 0.044 [2] dropping to 0.024K with the McMillan formula [3]. A BCS-calculation considering the full momentum dependent interaction gives T_c(max)=30K [3]. Initial studies using the same model gave expressions resulting in respectively T_c(max)=230K [4] and T_c(max)=1900K [5] for metallic hydrogen.

All these calculations used the same model for the phonon-mediated interaction, yet T_c(max) varied from 0.024K to 1900K. We think that 0.13K is accurate, but for this we had to solve the full momentum and frequency dependence of the interaction, which is non-standard. These results obviate the importance of improving methods for calculating T_c of superconducting materials.

[1] P. G. De Gennes, Superconductivity of Metals and Alloys, Benjamin (1966)
[2] L. Rademaker, D. van der Marel and Ch. Berthod, to be published.
[3] D. van der Marel and Ch. Berthod, Newton 1, 100002 (2025); D. van der Marel, Physica C 632, 1354688 (2025).
[4] D. A. Kirzhnits, ZhETF Pis. Red. 9, 360 (1969)
[5] V. L. Ginzburg and D. A. Kirzhnits, Nature. 220, 148 (1968)