## 2 Theory
For two-electron systems, one should also add the fourth term in Equation (14), which is the repulsive potential between two electrons. In addition, for two-nuclei systems, the repulsive potential between two-nuclei should be computed considering the fifth term in Equation (14). For the matrix elements of the kinetic energy of electrons in Equation (14) on the basis of a \(3n\)-dimensional CSs grid, by employing Equations (2) and (3), it is easy to obtain  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]3  
where \(j\) is the dimension number. The matrix elements of electron-nuclear Coulombic potentials in Equation (14) are also achieved by the following [31; 37]:  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]4  
where \(i\) and \(j\) are the index numbers of electrons and nuclei, respectively, and  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]5The matrix elements of electron-electron Coulombic potentials in Equation (14) are also computed by the following [31; 37]:  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]6  
where  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]7  
As is evident from Equations (16) and (18), one of the most important features of CSs is that they remove the singularity of Coulombic potentials and replace it with the complex error function (\(erf\)). In this work, we assume that the external laser field is linearly polarized along the \(z\)-axis and that the shape of the electric field is given by  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]8  
In Equation (20), \(E_{0}\) is the maximum amplitude of the laser field, \(\omega\) is the angular field frequency, and \(A_{env}(t)=\sin^{2}(\frac{t}{\tau}\pi)\) is the envelope with the full width at half maximum (FWHM) duration of \(\tau\). In the presence of an external laser field, for the matrix elements of the external laser potential on the basis of a SCS grid, we can easily obtain  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]9  
In order to compute the time-dependent expectation value of any observable \(O\) on the basis of a static grid of CSs, by employing the identity operator of CSs from Equation (11), one can write  
\[\hat{q}=\frac{-\gamma^{1/2}}{\sqrt{2}}\Big{(}\hat{a}^{\dagger}+\hat{a}\Big{)}, \quad\hat{p}=i\hbar\frac{\gamma^{1/2}}{\sqrt{2}}\Big{(}\hat{a}^{\dagger}+\hat{ a}\Big{)}. \tag{3}\]0  
Finally, by employing Equations (8) and (9) in Equation (22), we obtain  
\[\hat{q}=\frac{-\gamma^{1/2}}{\sqrt{2}}\Big{(}\hat{a}^{\dagger}+\hat{a}\Big{)}, \quad\hat{p}=i\hbar\frac{\gamma^{1/2}}{\sqrt{2}}\Big{(}\hat{a}^{\dagger}+\hat{ a}\Big{)}. \tag{3}\]1  
In order to obtain the ground state of the system, one must propagate Equation (13) in imaginary time until the expectation value of the Hamiltonian of the systems (from Equation (23)) converges to the lowest accessible value [31; 38]. In this part, there is no external laser field in the system. Moreover, obtaining the upper electronic states of the system is possible by employing the Gram-Schmidt algorithm on the basis of CSs [32]. For a two-nuclei system, propagation of the TDSE in imaginary time for a constant inter-nuclear distance would lead to the ground state of the system at that specific inter-nuclear distance. When implementing this approach for two-nuclei systems, it is necessary to repeat this process for a sufficient number of inter-nuclear distances to gain the potential energy curve of different electronic states of the system [32]. Here, we introduce another approach based on the SCS method that simulates the electronic-state potential energy curves of a two-nuclei system in the absence or presence of an ultra-short laser field. This approach only needs the electronic states of the system for an initial inter-nuclear distance, where the nucleus-nucleus force is strong enough to dissociate the two nuclei.