Application of the Cluster Variation Method to thes−dInteraction Hamiltonian

The ground-state energy of an $s-d$ interacting system is calculated through the cluster variation technique. This method has been slightly reformulated so that only the expectation values of the problem appear as unknowns. In order to treat the $s-d$ exchange Hamiltonian without decoupling, up to three-particle density matrices are retained in the entropy expansion. Minimization of the free energy leads to equations that can be solved for the expectation values at zero temperature. The ground-state energy lowering is calculated to be $W_0=-2N\left(0\right)\mathrm{}{D}^2 \exp -\left[\frac{4N}{3\mu \mathrm{JN}\left(0\right)\mathrm{}}\right]$, where $\mu =-1\mathrm{}$ for antiferromagnetic coupling ($J<\mathrm{}0\mathrm{}$), and $\mu =\frac{1}{3}$ for ferromagnetic coupling ($J>\mathrm{}0\mathrm{}$).