Numerical estimate of the anharmonic contribution to thermal expansion and free energy to orderλ4

Thermal-expansion contributions to the free energy and heat capacity of an anharmonic crystal have been estimated. It has been found that the perturbation expansion for the free energy converges faster by taking into consideration the thermal-expansion contribution. Due to thermal expansion, the $T^2$ coefficient of the heat capacity at constant volume is reduced to about 50% of its explicit anharmonic contributions. The expressions for thermal expansion have also been obtained to $O\left({\lambda }^3\right)$. The contribution of $O\left({\lambda }^3\right)$ has been found to enhance the lower-order results by about (4-7)% near room temperature.