## 3 Results and Discussion
### Computational Calculations
#### 3.4.1 The [CuII(\(\beta\)-CD)]\({}^{2+}\) Complex
The rim diameters of the \(\beta\)-CD in the optimized complex are measured according to the scheme shown in Figure 5A. For the atoms on the \(\beta\)-CD, the positions of the O\({}_{2}\)/O\({}_{3}\)/C\({}_{6}\) atoms are determined by the sugar ring while others are more flexible due to the rotations around C\({}_{2}\)\(-\)O\({}_{2}\), C\({}_{3}\)\(-\)O\({}_{3}\), and C\({}_{5}\)\(-\)C\({}_{6}\) bonds (Figure 5B). Therefore, the former three atoms can be used to reveal the rim size. It is known that the entrance size of the small rim of \(\beta\)-CD is defined by the atom-to-atom distances (R values) between C\({}_{6}\) on opposite rings. The secondary hydroxyl groups (O\({}_{2}\) and O\({}_{3}\) oxygens) form the bigger rim. The maximum and minimum values are labeled Rs (max), Rs (min), R\({}_{\rm L}\)(max) and R\({}_{\rm L}\)(min)respectively. For the small rim, seven C\({}_{6}\)\(\cdots\)C\({}_{6}\)\({}^{\prime}\) distances (C\({}_{6}\) in glucose unit 1 between C\({}_{6}\) in glucose unit 4,5,C\({}_{6}\) in glucose unit 2 between C\({}_{6}\) in glucose unit 5,6, C\({}_{6}\) in glucose unit 3 between C\({}_{6}\) in glucose unit 6,7 and C\({}_{6}\) in glucose unit 4 between C\({}_{6}\) in glucose unit 7) are measured. For the large rim, a set of three cross-ring distances are measured. For O\({}_{3}\) in glucose unit 1, the three measured distances are O\({}_{3}\) in glucose unit 1 between O\({}_{3}\) in glucose unit 4 and 5, together with the O\({}_{2}\) in glucose unit 4; for O\({}_{2}\) in glucose 1, they are O\({}_{2}\) in glucose unit 1 between O\({}_{2}\) in glucose unit 4 and 5, together with the O\({}_{3}\) in glucose unit 5. The maximum and minimum of all cross-ring distances calculated are recorded respectively. They reflect how much the upper/lower rim is opened up and their difference value(\(\Delta\)R) give us indication to the shape of the entrance. A bigger \(\Delta\)R refers to a more elliptical entrance while a smaller one corresponds to a rounder shape [27; 28]. Figure 4: Ball-and-stick model of optimized \(\beta\)-CD/Cu complexes with side and bottom views and its energy. (**A**): The initial structure when the copper ion is in the center of the small rim of the \(\beta\)-CD; (**B**): The optimized structure when the copper ion is in the center of the small rim of the \(\beta\)-CD. (**C**): The initial structure when the copper ion is in the edge of the small rim of the \(\beta\)-CD; (**D**): The optimized structure when the copper ion is in the edge of the small rim of the \(\beta\)-CD. Color coding: copper, ice blue; carbon, gray; oxygen, light red; hydrogen, white. The bond length (Å) was estimated for three O atoms. The large rim and the small rim of CD are both considered to be a possible binding site. Table 2 shows that, when the copper ion is added, the large rim diameters of \(\beta\)-CD change from 11.11 A, 10.25 A to 10.09 A, 9.82 A respectively, which is in accordance with the formation of Cu-O bond. The Rs(min) increases correspondingly, indicating that the small rim becomes bigger. Furthermore, the \(\Delta\)R of the \(\beta\)-CD/Cu complex become smaller in the existence of copper ion, which indicates that the \(\beta\)-CD entrance becomes rounder to accommodate small-molecule drugs more easily. In conclusion, we can see that the copper ion can form coordination bonds with oxygen atoms, changing the rim size of the \(\beta\)-CD. It has been reported that metal cations can assist host-guest interactions [27]. [PERSON] introduced Fe\({}^{2+}\) and Mg\({}^{2+}\) to enhance the detection of \(\beta\)-CD/toluene complexes. Using electrospray mass spectroscopy, the existence of ternary complexes (dication/\(\beta\)-CD/toluene) has been proved [28]. Therefore, it can be indicated that all these changes seem to make preparation for the binding other molecules.