The approximate method to treat the practical quantum anharmonic solids proposed by Hardy. Lacks and Shukla is reformulated with explicit physical meanings. It is shown that the quantum effect is important at low temperature, it at be treated in the harmonic framework; and the anharmonic effect is important at high temperature and tends to zero at low temperature, it can be treated by using a classical approximation. The alternative formulation is easier for various applications, and is applied to a Debye-Gruneisen aolid with the generalized Lennard-Jones intermolecular interaction. The expressions for the Debye temperature and Grunelsen parameter as a function of volume are analytically derived. The analytic equation of state is applied to predict the thermodynamic properties of solid xenon at normal pressure with the nearest-neighbour Lennard-Jones interaction, and is further applied to research the properties of solid xenon and krypton at high pressure by using an all-neighbour Lennard-Jones interaction. The theoretical results are in agreement with the experiments.

The cell-cluster expansion theory is analytically expanded with the anharmonic and the lowest order correlation effects taken into account. The thermodynamic quantities for the all-neighbor and the nearest-neighbor Lennard-Jones models are calculated. The results show that the agreement of the analytic approximate theory with the complete one and the computer simulation is quite good. The comparison also shows that the contributions of the non-nearest neighbors to the thermal parts of thermodynamic quantities are larger than the second-order correlation modification. The ignorance of second-order correlation modification is a reasonable and practically convenient approximation. A simple method is proposed to extend the analytic approximate theory to the quantum case. Considering the quantum effect is remarkable only at low temperature, and tends zero at high temperature, the potential energy of an atom is harmonically expanded to consider the quantum effect of solids within the harmonic oscillator framework. The anharmonic effect is remarkable only at high temperature, and tends zero at low temperature, it is disposed using a classical approximation. The results for the quantum theory are in good agreement with the experimental data of solid xenon.

In this letter, it is pointed out that an ideal universal equation of state (EOS) of solids should have four merits. The EOS corresponding to the generalized Lennard-Jones (GLJ) potential is derived. It is pointed out that the GLJ EOS is not volume analytic, as the exponents contained in the potential function take arbitrary values. On making the exponents satisfy a relationship, the GLJ EOS becomes volume analytic with two parameters (mGLJ EOS), and has all four merits. By applying six EOSs in investigating 50 materials, it is shown that the mGLJ EOS gives the best results.