Using models simulating the environment of two distinct adsorption sites of $H_2$ in metal-organic framework-5 (MOF-5), binding energies of $H_2$ to MOF-5 were evaluated at the MP2 and CCSD(T) level. For organic linker section modeled as dilithium 1,4-benzenedicarboxylate ($C_6H_4(COO)_2Li_2$), the MP2 and CCSD(T) basis set limit binding energies are estimated to be 5.1 and 4.4 kJ/mol, respectively. For metal oxide cluster section modeled as $Zn_4O(CO_2H)_6$, while the MP2 basis set limit binding energy estimate amounts to 5.4 kJ/mol, CCSD(T) correction to the MP2 results is shown to be insignificant with basis sets of small size. Substitution of benzene ring with pyrazine ring in the model for the organic linker section in MOF-5 is shown to decrease the $H_2$ binding energy noticeably at both the MP2 and CCSD(T) level, in contrast to the previous study based on DFT calculation results which manifested substantial increase of $H_2$ binding energies upon substitution of benzene ring with pyrazine ring in the similar model.