Crystallography is the principal method used to determine the structure of a ribosome, and consequently for understanding its functions, including formation of the peptide bond by ribozyme catalysis, and the decoding of the genetic code [1][2][3][4][5]. As shown in Figure 16.1, the ribosome is made of two subunits. It was found that the mRNA is decoded at the small subunit. The peptide bond is formed on the larger subunit within a cavity, hosting the peptidyl transferase center (PTC), composed mainly of ribosomal RNA [1][2][3][4][5][6][7][8].Of importance to the quantum calculations reviewed in this chapter, a region of pseudo twofold symmetry, which was detected in all known ribosome structures in and around the PTC [1,3b,3c,3d,3e], is associated with the translocation of the aminoacylated tRNA through the ribosome, as peptide bond formation occurs [2, 3], navigated by the ribosome architecture ( Figure 16.2). The nascent proteins move out of the ribosome via an exit tunnel whose opening lies adjacent to the PTC and receives thereby each successive peptide bond as the protein elongates. Thus the architecture of the ribosome is consistent with the requirements of peptide bond catalysis and protein formation [2, 3,5,9,10].Given the structural architecture of the ribosome, quantum crystallography (QCr) [11] may be applied to study the transition state (TS) for peptide bond formation. (The foundations and applications of QCr are reviewed in Chapter 1.) QCr combines crystallographic structural information with quantum mechanical theory. This facilitates theoretical calculations and adds an energetic aspect to crystallography. The crystallographic structure is a starting point, constraint and anchor for the quantum calculations. In QCr the molecular system is mathematically divided into computationally tractable pieces. Subsequently, this may be followed by a quantum investigation of their mutual interactions, and thus in a step-by-step manner one may rebuild the entire quantum mechanism as a whole. This approach has been applied to the investigation of the peptide bond TS. The first step here was to Quantum Biochemistry. Edited by Chérif F. Matta