It remains a general challenge to computationally design optimal catalytic structures based on earth-abundant metals for hydrogenation. Here, we demonstrate an effective computational approach based on inverse molecular design to deterministically design optimal catalytic sites on the Cu(100) surface through the doping of Fe and/or Zn, and a stable Zn-doped Cu(100) surface was found with minimal binding energy to H atoms. By the calculations at the level of density functional theory, the optimized catalyst sites are verified to be valid on the Cu(100) surface in an infinite periodic system. We analyze the electronic structure cause of the optimal binding sites using the analysis of the density of states. In addition, we use a Cu29Zn3 atomic cluster, where such an optimum catalytic site is valid on the Cu(100) surface, to understand the role of doped Zn atoms on lowering the H atom binding energy. We found that in the atomic cluster, the atomic orbitals of surface Zn-atoms show less participation in the binding of H atoms, compared to the atomic orbitals of surface Cu atoms. Our study provides valuable chemistry insights on designing catalytic structures using earth-abundant metals, and it may lead to the development of novel Cu-based earth-abundant alloys in bulk, nanoparticles, atomic clusters, or single-atom catalysts for important catalytic applications such as lignin degradation or CO2 conversion.