**1. Introduction**

The development of new technologies and experimental techniques has triggered intensive theoretical studies on modeling spatially confined quantum systems [1,2] and also extreme-high-pressure plasmas [3] like in stellar envelopes [4]. The thermodynamics of high-density hydrogen plasmas has been deeply investigated [5–9], due to the necessity of properly accounting for the effects of the multi-body interaction and in principle requiring the reformulation of the statistical mechanics in terms of a global Hamiltonian for the whole gas, instead of the usual separable form of non-interacting chemical species characterized through internal and translational partition functions. The non-ideality also affects the transport properties and in the case of dense, non-ideal, weakly-ionized Debye hydrogen plasma, the electrical conductivity in the non-metal-to-metal transition region at 150 GPa has been measured [10].

The investigation of the thermodynamic and transport properties of highly-dense hydrogen (and its isotopes) and helium plasmas is in fact relevant to many different fields, from astrophysics, for applications to low mass stars and giant planets [11], to inertial confinement fusion for the understanding of the ignition phase. Moreover, hot dense hydrogen and deuterium plasmas can be generated in a laboratory with shock compression, allowing the experimental accurate determination of the molecular-to-atomic transition along the principal Hugoniot to be compared with theoretical first-principle results [12].

It is also worth noting that atomic properties (level ensemble, electrical properties, static polarizability and hyperpolarizability [13–15] and optical oscillator strengths [16]) and the dynamics of collisions (electron impact excitation and ionization [17], symmetric charge exchange [18–20]) change in high-density regimes and are the subject in recent years of an intense investigation focused on the atomic hydrogen system.

In this paper, the thermodynamic properties and the electrical conductivity of weakly non-ideal, high-density partially ionized hydrogen plasma are investigated, accounting for quantum effects due to the change in the energy spectrum of atomic hydrogen when the electron-proton interaction is considered embedded in the surrounding particles. High-density conditions were simulated assuming atomic hydrogen described by a static screened Coulomb potential.

The Debye-Hückel or Yukawa potential, derived from the linearization of the exponential in the Poisson–Boltzmann equation [21,22], is considered suitable for the description of weakly-coupled plasmas, i.e., when the coupling parameter Γ = 1/(*akBTe*) ≤ 1, where *a* = [3/(<sup>4</sup>*π Ne*)]1/3 and *Ne* is the free electron density, and has been used in the literature for the estimation of the effects on collision processes [17,19,23–27]. The conditions explored in the present paper, the electron density ranging from 10<sup>16</sup> to 10<sup>23</sup> cm<sup>−</sup><sup>3</sup> and the temperature from 10<sup>4</sup> to 5 10<sup>4</sup> K, are compatible with weak coupling up to *ne*=1022, while for higher densities the value of Γ is greater than the unity and in principle would require a quantum approach to properly treat the interaction in these strongly-coupled plasmas. In fact, the chemical picture of the interaction offered by the Yukawa potential fails in a strongly correlated quantum regime, where other effects need to be accounted for, such as the ion-ion correlation, the electron exchange, the consistent statistics for electrons and therefore the accurate ab initial molecular dynamics method has to be resorted to [28–30]. Another important issue in both weakly- and strongly-coupled plasmas and neglected in this paper is the dynamical nature of screening, affecting the interaction potential between electrons and ions, and in turn, the transport properties of the plasma and the dynamics of elastic and reactive collisions [21,31,32]. In fact, plasma density fluctuations, due to inter-particle correlation in dense plasmas, produce time-dependent effects in the interaction of electrons and ions, due to the polarization induced by the electron on the surrounding plasma particles, that critically depends on the ratio between the electron velocity and its thermal velocity. The effect of dynamic screening on scattering processes in weakly-coupled plasmas has been investigated [25,33,34], showing that the use of static screening overestimates the shielding, therefore, we would expect an increase of the elastic transport cross-sections reducing the electrical and thermal conductivities.

## **2. Results**
