Here, a first-order LLPT for nitrogen-oxygen (N-O) mixtures is investigated via extensive ab initio molecular dynamics simulations. However, the physical understanding of LLPT in simple liquid mixtures has remained elusive. The first-order liquid-liquid phase transition (LLPT) describes the counterintuitive nature between two distinct liquids in a single system. Finally, a phase diagram of krypton is given, which provides a clear picture for understanding the thermophysical behavior of krypton in a wider temperature-pressure range. Our evaluation of the Karasiev-Sjostrom-Dufty-Trickey free-energy functional experimentally validates the XC thermal effect in the WDM regime, verifies the previous predictions, and sheds light on a direction for future theoretical efforts. However, the incorporation of long-range interactions into the SCAN (SCAN+rVV10 XC functional) results in a noticeably stiffer prediction due to an overestimation of the density and internal energy of the system at low densities and temperatures. It is found that the experimental data are better reproduced by the strongly constrained and appropriately normed (SCAN) XC functional compared to the conventional Perdew-Burke-Ernzerhof and Van der Waals (vdW) DF1 functionals, elucidating that the introduction of the kinetic energy density and the intermediate-range vdW interaction is decisive. The equation of states of krypton up to 155 GPa and 45 000 K, which ranges from an initial dense gaseous state up to the insulator-metal transition regime, were determined accurately. Motivated by the poor understanding of the applicability of new exchange-correlation (XC) functionals to warm dense matter (WDM), we designed and performed multiple-shock reverberation compression experiments on dense krypton to evaluate explicitly the implications of recently derived XC functionals. These results will provide an instructive basis for the experimental investigations of rare gases over a wide T-P range. With the help of simulation results and experimental data, a comprehensive phase diagram for krypton is constructed by using the solid-fluid and insulator-metal fluid phase boundaries, which fills the gap of the experimental work. The calculated electrical conductivities confirm that the metallization transition occurs at about 60 GPa and 17.5 kK along the principal Hugoniot. The QMD-simulated results of the ionic structures and electronic properties imply the occurrence of two kinds of phase transitions, including transition from a solidlike to fluid state and that from an insulator to conductive fluid in this T-P regime. It is found that, within the regime of the current density (ρ) and temperature (T), sound velocity can effectively discriminate differences between different theoretical models, and therefore it is more suitable as a benchmark to verify the practicability of models. The shock wave experimental data are used to validate the present theoretical models. Extensive quantum molecular dynamics (QMD) simulations are performed to determine the equation of state, sound velocity, and phase diagram of middle-Z krypton in a warm dense regime where the pressure (P) is up to 300 GPa and the temperature is up to 60 kK.
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