Internal energy of high density Hydrogen: Analytical approximations compared with path integral Monte Carlo calculations

The internal energy of high-density hydrogen plasmas in the temperature range $T = 10,000 ... 50,000 K$ is calculated by two different analytical approximation schemes (method of effective ion-ion interaction potential - EIIP and Pad\'e approach within the chemical picture - PACH) and compared with path integral Monte Carlo results. Reasonable agreement between the results obtained from the three independent calculations is found, the reasons for still existing differences is investigated. Interesting high density phenomena such as the formation of clusters and the onset of crystallization are discussed.