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.