## 3 Calculations and Results  
In this report, the implemented SCS grid, which consisted of 1000 CSs in the inner box and 500 CSs in the outer box, was generated using the Gaussian distribution function with \(\gamma=0.7\) and compression parameter [31] for the inner box of 0.9 and for the outer box of 1.0. CSs in the inner box were randomly distributed in the phase space around the origin in the \(x\), \(y\), \(p_{x}\), \(p_{y}\), and \(p_{z}\) directions and between (\(-6\) a.u., \(6\) a.u.) in the \(z\) direction. CSs in the outer box were randomly distributed in the phase space between (\(-2.5\) a.u., \(2.5\) a.u.) in the \(x\), \(y\), \(p_{x}\), and \(p_{y}\) directions; between (\(-10\) a.u., \(10\) a.u.) in the \(z\) direction; and between (\(-5\) a.u., \(5\) a.u.) in the \(p_{z}\) direction. We applied the classical nuclear dynamics approach to achieve the potential energy curves of the ground state and the first excited state of H\({}_{2}^{+}\). At first, we computed the ground state of the system in an initial inter-nuclear distance (\(R_{12}=1.0\) a.u.) by propagating the TDSE in imaginary time over the whole static grid. Then, to compute the first excited state of the system in the same initial inter-nuclear distance, we have employed the Gram-Schmidt algorithm [32]. Having computed the ground state and the first excited state of the system in an initial inter-nuclear distance, we propagated each of these electronic states in real time in the absence of any external field by considering classical nuclear dynamics. The initial velocity of the two nuclei was set to zero. In this way, the potential energy curves of the ground state and the first excited state of the system were calculated. The simulation results for the dynamic nucleus (DN) approach are plotted in Figure 1 and compared to the results from the static nucleus (SN) approach [32] and to the exact values [39]. It is evident from Figure 1 that the DN results had a good agreement with the exact values [39]. In Figure 1, for the ground state, the computation speed for the short inter-nuclear distances (less than \(1.5\) a.u.) was high in that the nuclear dynamics at short inter-nuclear distances was fast. As the inter-nuclear distance increased (particularly to greater than \(4\) a.u. ), the nuclear dynamics (and consequently the computation speed) became slower. For the next round of our investigation, the ground state of H\({}_{2}^{+}\) at \(R_{12}=1.45\) a.u. was introduced to a five-cycle ultra-short attosecond laser pulse using a wavelength of \(\lambda=70\) nm and different intensities. Attosecond pulses are needed in such investigations in that the electronic dynamics typically takes place on sub-femtosecond time scales. The shape of such laser pulses for the intensity of \(10^{14}\) W/cm\({}^{2}\) is plotted in Figure 2. The corresponding energy for a single photon excitation (17.712 eV = 0.6509 a.u.) could be high enough for exciting the ground state of the system to the first excited state at \(R_{12}=1.45\) a.u. Figure 1: Potential energy curve of the ground state and the first excited state of H\({}_{2}^{+}\) computed by employing the new dynamic nucleus (DN) approach. DN results are compared to the static nucleus (SN) approach results [32] (which are adopted with permission from the Journal of Computational Chemistry, published by John Wiley and Sons, 2018) and to the exact values [39] (which are adopted with permission from Atomic Data and Nuclear Data Tables, published by Elsevier, 1970.) Figure 2: The implemented laser pulse was a five-cycle attosecond laser pulse with a wavelength of 70 nm and intensity of \(10^{14}\) W/cm\({}^{2}\). As can be seen from Figure 3, we tuned our simulation in such a way that when the laser field reached its maximum amplitude, the inter-nuclear distance of H\({}_{2}^{+}\) reached \(R_{12}=1.45\) a.u. in the ground state. We repeated the simulation with different laser intensities (from \(0.1\times 10^{14}\) to \(2.0\times 10^{14}\) W/cm\({}^{2}\)).