## 3 Results and Discussion
### Coulomb Explosion of the Double-Charged Molecular Ion CHBrCl\({}_{2}\)\({}^{2+}\)  
With the increase of laser intensity, the CHBrCl\({}_{2}\) molecules can be ionized to the multi-charged molecular ions, then dissociate into fragment ions with higher KERs by Coulomb repulsive force. In the two-body Coulomb explosion process, the kinetic energy of a pair of fragment ions should satisfy Equation (4), where X and Y denote a pair of fragment ions, M is the mass of fragment ions, and p and q are the charge numbers of the ions:  
\[\text{KERs}(\text{X}_{\text{p+}})/\text{KERs}(\text{Y}_{\text{q+}})=\text{M}( \text{Y}_{\text{q+}})/\text{M}(\text{X}_{\text{p+}}), \tag{4}\]  
\[\text{Total KERs}=14.4\text{p}\text{q}/\text{r}, \tag{5}\]  
According to Equation (5), the Br\({}^{+}\)(1.45 eV) and CHCl\({}_{2}\)\({}^{+}\)(1.39 eV) ions can be attributed to the CE channel as follows:  
\[\text{Channel (3)}\qquad\text{CHBrCl}_{2}\rightarrow\text{CHBrCl}_{2}{}^{2+}+2 \text{e}^{-}\rightarrow\text{Br}^{+}+\text{CHCl}_{2}{}^{+}+2\text{e}^{-}\]  
The angular distributions of fragment ions further confirm the CE process. The angular distributions of ions Br\({}^{+}\) and CHCl\({}_{2}\)\({}^{+}\) are shown in Figure 6. They have the same angular distribution, which validates that the above dissociation channel (3) belongs to the CE process, along with the C-Br bond. In the classic CE model, the total KERs of one CE channel are expressed as Equation (5), where r indicates the internuclear distance in A. Once the total KERs are measured, the critical distance (R\({}_{\text{C}}\)) of the CE channel can be determined. According to our experimental results, the CE of CHBrCl\({}_{2}\)\({}^{2+}\) started at R\({}_{\text{c}}\) = 5.07 A. The R\({}_{\text{c}}\) is far longer than the equilibrium internuclear distance (R\({}_{\text{e}}\) = 1.95 A), which is caused by the internal Coulomb force stretching the nuclear distance. For polyatomic molecular ion CHBrCl\({}_{2}\)\({}^{2+}\) involved in CE processes, is it appropriate to use Coulomb repulsion forces between two atoms? Can the real potential energy relationship of molecules changing along the chemical bond determine the release of kinetic energy? To answer these questions, we calculated the potential energy curves of the multi-charged molecular ion CHBrCl\({}_{2}\)\({}^{2+}\) along with the C-Br bond at the B3 LYP/6-31G\({}^{*}\) level. The Coulomb repulsive curve also has been shown in Figure 7. Here, the energy in  
Figure 6: The angular distribution of CHCl\({}_{2}\)\({}^{+}\) (blue solid circle) and Br\({}^{+}\) (red hollow circle) ions, where the double arrow represents the laser polarization direction. 10 A on the potential energy curve (PEC) was equal to that of the Coulomb repulsive curve (CRC) for convenience [26]. As shown in Figure 7, when the total KERs of the channel (3) was 2.84 eV, the calculated C-Br distance in CRC and PEC was 4.33 and 5.07 A, respectively. Why is the effective distance of C and Br atoms longer in Coulomb explosion? We believe that the reason is the shift of the effective charge centers in the polyatomic molecule systems. Next, we calculated the effective centers of double-charged molecular ion CHBrCl\({}_{2}\)\({}^{2+}\). Firstly, the space configuration and charge distribution of the parent ions were calculated, and then the charge center coordinates on the quasi-fragment ion were obtained according to Equation (6). After the charge center coordinates were obtained, the critical value of the effective Coulomb explosion of polyatomic molecular ions was obtained by using the Euclidean distance. The results are in good agreement with the calculation results, indicating that it is different from the traditional classical point charge model. Hence, the movement of the effective charge center caused by molecular configuration needs to be considered in the process of the Coulomb explosion of polyatomic molecules.