A simple analytic expression with high precision for the radial distribution function of hard spheres is proposed. The form of the expression has been carefully selected to combine the well-known Camahan-Starling equation of state in it and satisfy the limit condition at low density, its simplicity and precision is superior to the well-known Percus-Yevick expression. The coefficients contained in the expression have been determined by fitting the Monte Carlo data for the first coordination shell, and by fitting both the Monte Carlo data and the numerical results of the Percus-Yevick expression for the second coordination shell. The expression has been applied to develop simple analytic equations of state for the hard-core single, double Yukawa fluids, and the hard-core Yukawa mixtures. The comparisons show that the agreement of our model with the computer simulation data is slightly better than the mean spherical approximation and other analytic models.

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.

The analytic mean-field approach (AMFP) was applied to the FCC C60 solid. For the intermolecular forces the Girifalco potential has been utilized. The analytic expressions for the Helmholtz free energy, internal energy and equation of state have been derived. The numerical results of thermodynamic quantities are compared with the molecular dynamic (MD) simulations and the unsymmetrized selfconsistent field approach (CUSF) in the literature. It is shown that our AMFP results are in good agreement with the MD data both at low and high temperatures. The results of CUSF are in accordance with the AMFP at low temperature, but at high temperature the difference becomes prominent. Especially the AMFP predicted that the FCC C60 solid is stable upto 2202 K, the spinodal temperature, in good agreement with 2320K from the MD simulation. However, the CUST just gives 1916 K, a temperature evidently lower than the MD data. The AMFP qualifies as a useful approach that can reasonably consider the anharmonic effects at high temperature.