The cluster-variation method (CVM) is discussed in the thermodynamic limit of an infinitely extended lattice. The relationship between the variational principle for the free energy per lattice point as valid in the thermodynamic limit and the CVM formalism is established. It is proved that for suitably chosen hierarchies of CVM approximations the $n\mathrm{th}$ approximation to the free energy per lattice point $f_n$ underestimates the exact value $f$ and that $f_n\to f$ monotonically as $n\to \infty$. CVM approximations of this kind permit a particularly appealing interpretation of the entropy approximation involved. As a practical consequence, the results of this theoretical investigation suggest how to construct a "best sequence of approximations" that can serve as a basis for extrapolations.

• Received 27 January 1983

DOI:https://doi.org/10.1103/PhysRevB.27.6841