In this study, we investigated the interaction between the H2S ligand and the heme pocket of hemoglobin I (HbI) of Lucina pectinata for the wild-type protein; three known mutations where distal glutamine is replaced by hydrophobic valine (Gln64Val) and hydrophilic histidine in both protonation forms (Gln64Hisϵ and Gln64Hisδ); five known mutations of the so-called phenyl cage, replacing the hydrophobic phenylalanines Phe29 and Phe43 with tyrosine (Tyr), valine (Val), or leucine (Leu); and two additional mutations, Phe68Tyr and Phe68Val, in order to complement previous studies with new insights about the binding mechanism at the molecular level. A particular focus was on the intrinsic strengths of the chemical bonds involved, utilizing local vibrational force constants based on combined quantum mechanical-molecular mechanical calculations. Wild-type protein and mutations clustered into two distinct groups: Group 1 protein systems with a proton acceptor in the distal protein pocket, close to one of the H2S bonds, and Group 2 protein systems without a hydrogen acceptor close by in the active site of the protein. According to our results, the interactions between H2S and HbI of Lucina pectinata involve two important elements, namely, binding of H2S to Fe of the heme group, followed by the proton transfer from the HS bond to the distal residue. The distal residue is additionally stabilized by a second proton transfer from the distal residue to COO- of the propionate group in heme. We could identify the FeS bond as a key player and discovered that the strength of this bond depends on two mutual factors, namely, the strength of the HS bond involved in the proton transfer and the electrostatic field of the protein pocket qualifying the FeS bond as a sensitive probe for monitoring changes in H2S ligation upon protein mutations. We hope our study will inspire and guide future experimental studies, targeting new promising mutations such as Phe68Tyr, Phe68Val, or Phe43Tyr/Phe68Val.