Hydrogen-rich compounds under extreme pressure have attracted significant attention as a platform for high-temperature superconductivity. While superconducting density functional theory (SCDFT) has been successfully applied to predicting Tc, a unified first-principles understanding of the fundamental superconducting length scales has remained a major challenge. In particular, the coherence length and magnetic penetration depth are key quantities governing pairing strength, phase stiffness, and electromagnetic response, yet their microscopic evaluation on equal footing with Tc has been lacking.

In this talk, we present a first-principles framework incorporating finite-momentum Cooper pairs within SCDFT, which enables a parameter-free and fully consistent determination of the coherence length, penetration depth, and Tc. By directly analyzing the momentum dependence of the superconducting gap and current, we establish a quantitative link between electronic structure and macroscopic superconducting properties.

Applying this framework to H3S under high pressure, we find a combination of a short coherence length and a relatively small penetration depth, placing H3S deep in the type-II regime. Moreover, the unified access to Tc, coherence length, and penetration depth allows us to construct the Uemura relation entirely from first principles, demonstrating that hydride superconductors occupy a unique regime bridging conventional BCS systems and high-Tc materials. This work establishes superconducting length scales as predictive and design-relevant quantities in first-principles theory, opening a new route toward a microscopic understanding and the design of new hydride superconductors.