## 3 Calculations and Results
An interesting phenomenon that could happen at this point is the charge migration between the ground state and the first excited state of the system [8; 40]. The wave function of the resulting coherent superposition state that corresponds to a spatial displacement of the electronic charge is generally expressed by  
\[\Psi_{s}(\mathbf{r},t;\mathbf{R}_{1},\mathbf{R}_{2})=c_{g}(t)\exp\biggl{(}-i \frac{E_{g}(R_{12})t}{\hbar}\biggr{)}\Psi_{g}(\mathbf{r};\mathbf{R}_{1}, \mathbf{R}_{2})+c_{u}(t)\exp\biggl{(}-i\frac{E_{u}(R_{12})t}{\hbar}\biggr{)} \Psi_{u}(\mathbf{r};\mathbf{R}_{1},  
As can be seen from Figure 4, by employing Equation (50), we computed the period of charge migration between the two nuclei in H\({}_{2}^{+}\) in terms of the inter-nuclear distance. This figure shows that greater inter-nuclear distances corresponded to longer charge migration times between the two nuclei. In Figure 4, it can be seen that the charge migration period in inter-nuclear distances between 1 and 2 a.u. was relatively low (about 200-300 as) and increased as the inter-nuclear distance grew. From Figure 4, it can be also conceived that at distances greater than \(R_{12}=6.0\) a.u., the electron density was practically localized on one nucleus.  
In order to compute the population of each electronic state \(\Psi_{g/u}\) in the coherent superposition state \(\Psi_{s}\), implementing the identity operator of CSs (Equation (11)), one should compute  
\[\left|\left\langle\Psi_{g/u}\middle|\Psi_{s}\right\rangle\right|^{2}=\left| \sum_{kl}\left\langle\Psi_{g/u}\middle|Z_{k}\right\rangle(\Omega^{-1})_{kl} \left\langle Z_{l}\middle|\Psi_{s}\right\rangle\right|^{2}. \tag{51}\]  
Applying Equations (8) and (9), we reach  
\[\left|\left\langle\Psi_{g/u}\middle|\Psi_{s}\right\rangle\right|^{2}=\left| \sum_{k}C_{g/u_{k}}^{*}D_{sk}\right|^{2}. \tag{52}\]  
From the population results in Table 1, which were computed (at \(t=270\) a.u.) using Equation (52), one can deduce that as the intensity of a laser increases, the population of the ground state in the coherent superposition state becomes lower and the population of the excited states becomes higher. In agreement with this deduction, it can be also seen from Figure 3 that coherent superposition states created by lasers with a lower intensity are closer to the ground state. A higher laser intensity would also lead to a coherent superposition state with more contributors. For example, the coherent superposition state created by using a laser field with an intensity of \(2.0\times 10^{14}\)W/cm\({}^{2}\) (in Figure 3) had more contributors. Table 1 and Figure 3 demonstrate that after exposing the ground state of the system  
Figure 4: The charge migration period computed by employing Equation (50) for different inter-nuclear distances in H\({}_{2}^{+}\) in the absence of an ultra-short intense laser field. to an attosecond pulse, some of the population was promoted to the excited states. As the intensity of the attosecond laser field was increased, a larger population was transferred to the excited states. Figure 5 shows the expectation value of the electron coordinate along the \(z\)-axis experiencing ultra-short intense laser pulses with four different intensities. One can see that the amplitude of the coordinate expectation value was larger for higher intensities. However, until 2.0 fs, the period seemed to be the same for all intensities. As the dissociation rate of the two nuclei increased, the period became longer for higher intensities (above \(t=2.0\) fs). In addition, we calculated the change rate of the inter-nuclear distance for the ground state and the first excited state in the absence of a laser field and for the ground state induced by ultra-short laser pulses with different intensities. The results, depicted in Figure 6, show that the lowest (highest) rates of dissociation corresponded to the ground (first excited) state in the absence of a laser field.