# Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method
## Theory
### Channel Orbital-Based TDCIS in the Velocity Gauge with Rotated Orbitals  
The gauge dependence of the LG and VG treatments, Equations (28) and (30), results from the fact that the ansatz of Equations (19) and (24), both using fixed orbitals, cannot be connected with the transformation, Equation (16), as is generally the case for truncated CI expansion using fixed orbitals. For a method to be gauge invariant, the underlying Lagrangian in LG and VG cases should be numerically the same when evaluated with the solution of respective EOMs, which does not hold in the present case, \(L_{\rm L}(t)\
eq L_{\rm V}(t)\), with Equations (21) and (25). Thus we _define_ the total wavefunction \(|\Psi_{\rm V}^{\prime}(t)\rangle\), transformed from \(|\Psi_{\rm L}(t)\rangle\) to the velocity gauge, as  
\[|\Psi_{\rm V}^{\prime}(t)\rangle=\hat{U}(t)|\Psi_{\rm L}(t)\rangle=|\Phi^{ \prime}\rangle C_{0}(t)+\sum_{i}^{occ}\sum_{a}^{vir}|\Phi_{ia}^{\prime}\rangle C _{ia}(t), \tag{31}\]  
with \(|\Psi_{\rm L}(t)\rangle\) constructed with the solution of CI coefficient-based EOMs in the LG, Equation (22). Here \(|\Phi^{\prime}\rangle=\hat{U}(t)|\Phi\rangle\) and \(|\Phi_{ia}^{\prime}\rangle=\hat{U}(t)|\Phi_{ia}\rangle=\sum_{\sigma}\sigma_{a \sigma}^{\rm rt}\sigma_{i\sigma}^{\prime}|\Phi^{\prime}\rangle/\sqrt{2}\) are the reference and singly-excited CSF constructed with unitary rotated orbitals, i.e., \(|\phi_{p}^{\prime}\rangle=\hat{U}|\phi_{p}\rangle\) and \(\hat{c}_{p\sigma}^{\prime}=\hat{U}(t)\hat{c}_{p\sigma}\hat{U}^{-1}(t)\). It shouldbe noted that \(|\Psi^{\prime}_{\rm V}\rangle\) cannot be rewritten into the form of Equation (24) in general. Associated with this wavefunction, we consider the following Lagrangian,  
\[L^{\prime}_{\rm V}(t)=\langle\Psi^{\prime}_{\rm V}|(\hat{H}_{\rm V}-i\partial_{t })|\Psi^{\prime}_{\rm V}\rangle. \tag{32}\]  
The equivalence of this approach to the LG treatment is readily confirmed by seeing  
\[L^{\prime}_{\rm V}(t)=\langle\Psi_{\rm L}|\hat{U}^{-1}(\hat{H}_{\rm V}-i \partial_{t})\hat{U}|\Psi_{\rm L}\rangle=\langle\Psi_{\rm L}|(\hat{H}_{\rm L} -i\partial_{t})|\Psi_{\rm L}\rangle=L_{\rm L}(t). \tag{33}\]  
One may naively expect that \(L^{\prime}_{\rm V}\) of Equation (32), which differs from \(L_{\rm V}\) of Equation (25) only by the replacement of \(\Psi_{\rm V}\) with \(\Psi^{\prime}_{\rm V}\), leads to the EOMs of Equation (26) with \(D_{0}\), \(\{D_{ia}\}\),\(\{\phi_{p}\}\) replaced with \(C_{0}\), \(\{C_{ia}\}\),\(\{\phi^{\prime}_{p}\}\). This is not the case, however, due to the time dependence of the rotated CSFs, e.g., \(\langle\Phi^{\prime}|\partial_{t}|\Phi^{\prime}_{ia}\rangle=i\mathbf{E}(t)\cdot \langle\Phi^{\prime}|\hat{\mathbf{r}}|\Phi^{\prime}_{ia}\rangle\), and after extracting these time dependence, Equation (32) reads  
\[L^{\prime}_{\rm V}(t)=\langle\Psi^{\prime}_{\rm V}|\{\hat{H}_{\rm V}+\mathbf{E}(t) \cdot\hat{\mathbf{r}}-i\partial_{t}^{c}\}|\Psi^{\prime}_{\rm V}\rangle, \tag{34}\]  
where \(\partial_{t}^{c}\) time differentiates CI coefficients only. Now requiring \(\partial L^{\prime}_{\rm V}/\partial C^{*}_{0}=\partial L^{\prime}_{\rm V}/ \partial C^{*}_{ia}=0\), or equivalently, substituting the back transformation \(|\phi_{p}\rangle=\hat{U}^{-1}|\phi^{\prime}_{p}\rangle\) into Equation (22) derives  
\[i\partial_{t}C_{0} = \sqrt{2}\mathbf{E}\cdot\sum_{jb}\langle\phi^{\prime}_{j}|\hat{\mathbf{r} }|\phi^{\prime}_{b}\rangle C_{jb}, \tag{35a}\] \[i\partial_{t}C_{ia} = \langle\phi^{\prime}_{a}|\{\sum_{b}(\hat{F}^{\prime}_{i}+\mathbf{A} \cdot\hat{\mathbf{p}}+\mathbf{E}\cdot\hat{\mathbf{r}})|\phi^{\prime}_{b}\rangle C_{ib}+ \sqrt{2}\mathbf{E}\cdot\hat{\mathbf{r}}|\phi^{\prime}_{i}\rangle C_{0}\}-\mathbf{E}\sum_{ j}C_{ja}\cdot\langle\phi^{\prime}_{j}|\hat{\mathbf{r}}|\phi^{\prime}_{i}\rangle. \tag{35b}\]  
where \(\hat{F}^{\prime}_{i}\) is given by Equation (23) with \(\{\phi_{j}\}\) replaced with \(\{\phi^{\prime}_{j}\}\). Equation (35) are the CI coefficient-based TDCIS EOMs based on the Lagrangian of Equation (32).