Thermodynamic properties of permanent dipoles on a lattice

The thermodynamic variational approximation is applied to the Gaussian integral representation of the partition function of a system of permanent dipoles. Upper and lower bounds to the free energy are obtained, as well as an approximation to the Clausius-Mossotti function $\alpha$. The $\beta \to \infty$ limit of $\alpha$ as a function of $n$, the dimensionality of the dipoles, is given. For the case of $n=3\mathrm{}$, the result is found to lie between that obtained via the spherical-model approximation and that obtained via Padé-approximant methods.