First-principles estimation of the superconducting transition temperature of a metallic hydrogen liquid

We present a first-principles implementation of the stochastic path-integral approach proposed by Liu et al.~\cite{liu_superconducting_2020} for estimating the superconducting transition temperature ($T_c$) of a liquid. The implementation is based on the all-electron projector-augmented-wave (PAW) method. We generalize Liu et al.'s formalism to accommodate the pseudo-description of electron states in the PAW method. A formula for constructing the overlap operator of the PAW method is proposed to eliminate errors due to the incompleteness of a pseudo-basis set. We apply the implementation to estimate $T_c$'s of metallic hydrogen liquids. It confirms Liu et al.'s prediction that metallic hydrogen can form a superconducting liquid.