## 3 Results and Discussions  
We use the molecule H\({}_{2}^{+}\) as a benchmark system, which can be fully investigated [57] to describe the photoelectron imaging process of coherent electron dynamics. The molecule at equilibrium aligned along the \(z\)-axis is excited by a linearly polarized \(\lambda_{pu}=70\) nm XUV pump laser pulse with its field vector along the \(x\)-axis. Subsequently, a \(\lambda_{pr}=5\) nm soft X-ray attosecond probe pulse with its field vector polarized in the \((x,y)\) plane ionizes the excited molecule. For the processes by high-frequency pump-probe pulses, multi-photon ionization dominates with the Keldysh parameter \(\gamma\gg 1\)[9]. Since the ponderomotive energy \(U_{p}=E_{0}^{2}/4\omega_{pr}^{2}\) is very weak, the modification of the ionization potential by laser induced Stark shifts can be ignored as well. Moreover, the dipole approximation, in which the spatial dependence and magnetic component of the external field are neglected, remains valid. We first present the coherent excitation and charge migration by a pump pulse. With a linearly polarized pump pulse with its field vector polarized along \(x\)-axis, perpendicular to the molecular axis, a \(\sigma_{g}-\pi_{u}\) parallel resonant pump excitation occurs from the electronic ground \(1s\sigma_{g}\) state, \(\psi_{\sigma_{g}}(\mathbf{r})\) with the eigenenergy \(E_{\sigma_{g}}\) to the electronic excited \(2p\pi_{u}\) state, \(\psi_{\pi_{u}}(\mathbf{r})\) with the eigenenergy \(E_{\pi_{u}}\). In our numerical models, the energies for the \(1s\sigma_{g}\) and \(2p\pi_{u}\) states are, respectively, \(E_{\sigma_{g}}=-1.08\) a.u. and \(E_{\pi_{u}}=-0.43\) a.u. To induce molecular resonant excitation, the wavelength \(\lambda_{pu}=70\) nm (\(\omega_{pu}=0.65\) a.u.) of the pump pulse is required, where \(\omega_{pu}=E_{\pi_{u}}-E_{\sigma_{g}}=0.65\) a.u. A coherent superposition of the two electronic states is then created due to a strong charge-resonance excitation [58],  
\[\psi_{0}(\mathbf{r},t)=c_{\sigma_{g}}(t)\psi_{\sigma_{g}}(\mathbf{r})e^{-iE_{ \sigma_{g}}t}+c_{\pi_{u}}(t)\psi_{\pi_{u}}(\mathbf{r})e^{-iE_{\pi_{u}}t}. \tag{7}\]  
\(c_{\sigma_{g}}(t)\) and \(c_{\pi_{u}}(t)\) are the occupation coefficients. \(\psi_{\pi_{u}}(\mathbf{r})\) is the degenerate quantum electronic state. For the case excited by the linearly polarized 70 nm XUV laser pulse with its vector along the \(x\)-axis, the wavefunction of the degenerate excited state is given by [59],  
\[\psi_{\pi_{u}}(\mathbf{r})=[\psi_{\pi_{u}}^{+}(\mathbf{r})+\psi_{\pi_{u}}^{-} (\mathbf{r})]/\sqrt{2}. \tag{8}\]  
By describing the electronic motions in the cylindrical coordinate space with \(\mathbf{r}=(\rho,\theta,z)\) with \((x=\rho\cos\theta,y=\rho\sin\theta)\) and aligning the molecule along the \(z\)-axis, the wave function of the degenerate excited state can be written as  
\[\psi_{\pi_{u}}^{\pm}(\rho,z)=\bar{\psi}_{\pi_{u}}^{\pm}(\rho,z)\exp(\pm i \theta), \tag{9}\]  
where \(|\bar{\psi}_{\pi_{u}}^{\pm}(\rho,z)|^{2}=[|\psi_{\pi_{u}}^{x}(\rho,z)|^{2}+| \psi_{\pi_{u}}^{y}(\rho,z)|^{2}]/2\), and \(|\psi_{\pi_{u}}^{x}(\rho,z)|^{2}\) and \(|\psi_{\pi_{u}}^{y}(\rho,z)|^{2}\) are the degenerate real orbitals.