## 2 Theory  
On the basis of the SCS method, in order to simulate a single- or two-electron system, a static grid of three- or six-dimensional CSs, respectively, is constructed in a phase space using a Gaussian distribution function. Having generated the CSs grid, it remains constant throughout the whole simulation. To achieve this, for each dimension of every electron in the system, the same number of one-dimensional CSs is generated using\[z=\frac{\gamma^{1/2}}{\sqrt{2}}q+i\frac{-\gamma^{1/2}}{\sqrt{2}\hbar}p, \tag{1}\]  
where \(\gamma\) tunes the width of CSs in phase space. In Equation (1), \(q\) is the position and \(p\) is the momentum of the 1D CS. Using the fact that CSs are eigenkets of the annihilation operator and eigenbras of the creation operator:  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]  
it is easy to verify that these two operators are related to the position and momentum operators in each dimension in such a way that  
\[\hat{q}=\frac{-\gamma^{1/2}}{\sqrt{2}}\Big{(}\hat{a}^{\dagger}+\hat{a}\Big{)}, \quad\hat{p}=i\hbar\frac{\gamma^{1/2}}{\sqrt{2}}\Big{(}\hat{a}^{\dagger}+\hat{ a}\Big{)}. \tag{3}\]  
For one-electron systems, a set of 3D CSs is constructed from 1D CSs of each dimension of the electron using the tensor product  
\[\ket{Z}=\ket{z_{1}}\otimes\ket{z_{2}}\otimes\ket{z_{3}}=\ket{z_{1}z_{2}z_{3}}. \tag{4}\]  
For two-electron systems, two sets of 3D CSs corresponding to each electron construct a set of 6D CSs:  
\[\ket{Z}=\ket{z_{e_{1}}}\otimes\ket{z_{e_{2}}}=\ket{z_{1}z_{2}z_{3}z_{4}z_{5}z_ {6}}. \tag{5}\]  
For two-electron systems, as the system is fermionic, the CSs grid should be constructed in such a way that the total wave function of the system becomes anti-symmetric [29; 31; 32]. For example, in the ground state of a two-electron system, as the spin wave function is anti-symmetric, the spatial wave function should be symmetric. Because the SCS method deals with the spatial wave function, the static CSs grid should be symmetrized (or anti-symmetrized) in order to be capable of simulating the symmetric (or anti-symmetric) electronic states of a two-electron system [31; 32]. Three- or six-dimensional CSs in the static CSs grid are non-orthogonal and make an over-complete basis set:  
\[\bra{Z}Z^{\prime}=\sum_{j=1}^{3n}exp\left(-\frac{1}{2}\bigg{(}\ket{z_{j}}^{2} +\ket{z^{\prime}_{j}}^{2}\bigg{)}+z^{*}_{j}Z^{\prime}_{j}\right), \tag{6}\]  
where \(n\) is the number of electrons in the system. The wave function of a single- or two-electron system can be represented as a superposition of, respectively, \(N\) three- or six-dimensional CSs:  
\[\ket{\Psi}=\sum_{k=1}^{N}D_{k}\ket{Z_{k}}. \tag{7}\]  
In Equation (7), for \(D_{k}\) coefficients, we have  
\[D_{k}=\sum_{l=1}^{N}\Big{(}\Omega^{-1}\Big{)}_{kl}C_{l}, \tag{8}\]  
where  
\[C_{l}=\bra{Z_{l}}\Psi \tag{9}\]  
and \(\Omega^{-1}\) is the inverse of the overlap matrix \(\Omega\) with elements  
\[\Omega_{kl}=\bra{Z_{k}}Z_{l}. \tag{10}\]Applying the identity operator of CSs [27; 31]:  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]0  
to the TDSE, we obtain  
(12)  
Because, in the SCS method, CSs are not evaluated by time, we can easily bring them into the time derivative. Then by implementing Equations (8) and (9), we obtain  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]1  
where \(H\) is the Hamiltonian of the system. Considering no quantum dynamics for nuclei, the general Hamiltonian for a single- or two-electron system would be  
\[\hat{a}\ket{z}=z\ket{z},\quad\bra{z}\hat{a}^{\dagger}=\bra{z}z^{*}, \tag{2}\]2  
where \(n,m=1,2\) are the indexes of electrons and nuclei in the system, respectively. In Equation (14), the first term is for the kinetic energy of \(n\) electrons, and the second term is for the electron-nuclear Coulombic potentials. In the presence of an external laser field, considering the dipole moment approximation, the third term would be added to the first two terms.