Self-Consistent Perturbation Theory for Thermodynamics of Magnetic   Impurity Systems
  Integral equations for thermodynamic quantities are derived in the framework of the non-crossing approximation (NCA). Entropy and specific heat of 4f contribution are calculated without numerical differentiations of thermodynamic potential. The formulation is applied to systems such as PrFe4P12 with singlet-triplet crystalline electric field (CEF) levels. 
J. Otsuki, H. Kusunose and Y. Kuramoto