## 4 Discussion  
The observed spectral differences of the studied saccharides can be qualitatively explained in terms of molecular structure, H-bonding network, and the presence of water in the crystallographic structure. The pattern of absorption peaks looks similar for \(\beta\)LA and CBS (Figure 2), and it is possible to appreciate an overall shift towards the higher frequencies of the spectrum of CBS compared to that of \(\beta\)LA. The crystal structure of \(\beta\)LA is nearly isomorphous with CBS. However, the H-bonding scheme around the axial O(4) atom of \(\beta\)LA is different from that of CBS. In particular, the \(\beta\)LA structure presents just one of the three H-bonds of CBS [56,57]. The fact that in \(\beta\)LA, the O(4) atom of the galactose subunit is connected through a single H-bond with the O(3') atom of a neighboring glucose subunit can roughly explain the shift of THz peaks for \(\beta\)LA to lower frequencies. As shown in Section 2.8, of the highest intensity absorption peaks of \(\beta\)LA and CBS (at 1.83 and 2.07 THz, respectively) approximately 60% are due to intermolecular librations. The presence of fewer H-bonds gives a weaker overall force constant (\(K\)) of the overall chemical bonds that regulate these intermolecular librations of the molecules. Since the oscillating mass is the same for \(\beta\)LA and CBS (their molecular mass is \(M=342.297\) uma), according to the harmonic oscillator frequency formula \(\
u_{o}=\frac{1}{2\pi}\sqrt{\frac{K}{M}}\), a lower normal mode vibrational frequency \(\
u_{o}\) is expected. In order to estimate \(K\), we applied this formula to the 1.83 THz peak of \(\beta\)LA and the 2.07 THz peak of CBS, obtaining \(K\) values of approximately  
Figure 5: Mode analysis of the studied saccharides: (**a**) GLU, (**b**) GAL, (**c**) \(\alpha\)LM, (**d**) \(\alpha\)LA, (**e**) \(\beta\)LA, (**f**) CBS, and (**g**) CEL. Each panel is divided into three graphs that show for each mode the percentage contributions of the three intermolecular translations (top), the three intermolecular librations (middle), and the intramolecular vibrations (bottom). 75 and 96 N/m, respectively. The difference is approximately 21 N/m. Since CBS possesses two additional H-bonds, by dividing this value by two, we found \(K\simeq 10.5\) N/m, a value within the force constant range of the H-bond, from approximately 3 to 80 N/m depending on the molecule [58]. The THz spectra of CBS and CEL (Figure 2a and Table 2) appear to be very different: while CBS shows five evident peaks, CEL is characterized by two small intensity peaks at 2.15 and about 3 THz superimposed to the monotonic increase. Even if CBS is the repeating unit of CEL, the supramolecular structure of the two compounds is very different due to different intermolecular H-bond networks. The CEL supramolecular structure is characterized by a network of inter- and intramolecular H-bonds, which give great stiffness to CEL chains made up of hundreds to thousands of glucose units [59]. CBS consists of two glucose units linked by a \(\beta\)-(1,4\({}^{\prime}\)) glycosidic bond, and its structure is, therefore, more flexible than that of CEL. These structural differences may explain the presence of the several vibrational peaks in the CBS spectrum compared to that of CEL. GLU and GAL are stereoisomers of each other. The main structural difference is the opposite orientation of the OH groups at carbon atom C(4) in the pyranose ring. Despite this, GLU and GAL exhibit very different THz spectra. The rearrangement of the hydroxyl group changes the potential energy distribution, and this most likely causes differences in the intermolecular vibrational motions, giving very distinct THz spectra. A GLU unit linked to a GAL unit through \(\beta\)-(1,4) glycosidic linkage forms \(\alpha\)LA. If the THz spectra of GLU and GAL are compared with the THz spectrum of \(\alpha\)LA (Figure 3), very marked differences are evident.