# Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method
## Theory  
### System Hamiltonian and Gauge Transformation  
Let us consider an atom or a molecule consisting of \(N\) electrons interacting with an external laser field. In this work, we restrict our treatment in the clamped-nuclei approximation and the electron-laser interaction within the electric dipole approximation. Then the exact description of the system dynamics is given by the solution \(\Psi_{\mathrm{L}}(t)\) of TDSE,  
\[i\partial_{t}\Psi_{\mathrm{L}}(t)=H_{\mathrm{L}}(t)\Psi_{\mathrm{L}}(t), \tag{3}\]  
with the system Hamiltonian \(H_{\mathrm{L}}(t)=H_{0}+H_{\mathrm{L}}^{\mathrm{ext}}(t)\), where \(H_{0}\) is the field-free electronic Hamiltonian  
\[H_{0}=\sum_{k=1}^{N}h(\mathbf{r}_{k},\mathbf{p}_{k})+\sum_{k=1}^{N}\sum_{l>k}^{N}\frac {1}{|\mathbf{r}_{k}-\mathbf{r}_{l}|}, \tag{4}\]  
where \(\mathbf{r}_{k}\) and \(\mathbf{p}_{k}=-i\mathbf{\
abla}_{k}\) are the coordinate and canonical momentum of an electron, \(h(\mathbf{r},\mathbf{p})=\frac{1}{2}\mathbf{p}^{2}+v_{\mathrm{n}}(\mathbf{r})\), with \(v_{\mathrm{n}}\) being the electron-nucleus interaction. Here we are considering the LG treatment, where the electron-laser interaction \(H_{\mathrm{L}}^{\mathrm{ext}}\) is given by  
\[H_{\mathrm{L}}^{\mathrm{ext}}(t)=\mathbf{E}(t)\cdot\sum_{k=1}^{N}\mathbf{r}_{k}, \tag{5}\]  
where \(\mathbf{E}(t)\) is the laser electric field. As is well known, the system dynamics is _equivalently_ described in the VG, of which the wavefunction \(\Psi_{\mathrm{V}}\) is connected with the LG one through  
\[\Psi_{\mathrm{V}}(t)=U(t)\Psi_{\mathrm{L}}(t), \tag{6}\]  
with a unitary transformation  
\[U(t)=\exp\left[-i\sum_{k=1}^{N}\left\{\mathbf{A}(t)\cdot\mathbf{r}_{k}-\frac{1}{2} \int_{-\infty}^{t}dt^{\prime}|\mathbf{A}(t^{\prime})|^{2}\right\}\right], \tag{7}\]  
where \(\mathbf{A}(t)=-\int_{-\infty}^{t}\mathbf{E}(t^{\prime})dt^{\prime}\) is the vector potential, and we arbitrarily include the second term in the exponential, which is a \(c\)-number, to avoid appearance of terms proportional to \(|\mathbf{A}|^{2}\) in subsequent equations. Then we substitute \(\Psi_{\mathrm{L}}=U^{-1}\Psi_{\mathrm{V}}\) into the LG TDSE, Equation (3), use \(dU/dt=i\sum_{k=1}^{N}(\mathbf{E}\cdot\mathbf{r}_{k}+|\mathbf{A}|^{2}/2)U\), and note \(UP_{k}U^{-1}=\mathbf{p}_{k}+\mathbf{A}\) to derive the VG TDSE,  
\[i\partial_{t}\Psi_{\mathrm{V}}(t)=H_{\mathrm{V}}(t)\Psi_{\mathrm{V}}(t), \tag{8}\]  
with \(H_{\mathrm{V}}(t)=H_{0}+H_{\mathrm{V}}^{\mathrm{ext}}(t)\), and  
\[H_{\mathrm{V}}^{\mathrm{ext}}(t)=\mathbf{A}(t)\cdot\sum_{k=1}^{N}\mathbf{p}_{k}. \tag{9}\]One should carefully note that the present proof of equivalence of the LG and VG treatments, Equations (3) and (8), with the transformation of Equation (7), applies only to the exact solution of TDSE. See e.g., Ref. [45; 46; 47] for deeper discussions on the gauge transformation within TDSE, and Ref. [25] for the gauge invariance of TD-MCSCF methods. For a compact presentation of the many-electron theory, we rewrite the system Hamiltonian in the second quantization,  
\[\hat{H}_{\rm L}(t)=\hat{H}_{0}+\hat{H}_{\rm L}^{\rm ext}(t),\quad\hat{H}_{\rm V }(t)=\hat{H}_{0}+\hat{H}_{\rm V}^{\rm ext}(t), \tag{10}\]  
\[\hat{H}_{0}=\hat{h}+\frac{1}{2}\sum_{\sigma\tau}^{\uparrow\downarrow}\sum_{ pqrs}\langle pr|qs\rangle\hat{c}_{pc}^{\dagger}\hat{c}_{rr}^{\dagger}\hat{c}_{q \tau}\hat{c}_{s\sigma}, \tag{11}\]  
\[\hat{H}_{\rm L}^{\rm ext}(t)=\mathbf{E}(t)\cdot\hat{r},\quad\hat{H}_{\rm V}^{\rm ext }(t)=\mathbf{A}(t)\cdot\hat{p}, \tag{12}\]  
where \(\{\hat{c}_{pv}^{\dagger}\}\) and \(\{\hat{c}_{pv}\}\) are the creation and annihilation operators, respectively, for the set of spin-orbitals given as a direct product \(\{\phi_{p}\}\otimes\{s_{\uparrow},s_{\downarrow}\}\) of orthonormal spatial orbitals \(\{\phi_{p}\}\) and up-spin (down-spin) functions \(s_{\uparrow}\) (\(s_{\downarrow}\)).