# Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method
## Numerical Examples  
In this section, we numerically apply the channel orbital-based TDCIS method in the LG, VG, and rVG to the 1D model Helium atom, using the computational code developed by modifying an existing TDHF code used in our previous work [30; 48; 33]. The field-free electronic Hamiltonian is given by  
\[H_{0}=\sum_{k=1}^{2}\left\{-\frac{1}{2}\frac{\partial^{2}}{\partial z_{k}^{2}}- \frac{2}{\sqrt{z_{k}^{2}+1}}\right\}+\frac{1}{\sqrt{(z_{1}-z_{2})^{2}+1}}, \tag{42}\]  
for two electronic coordinates \(z_{1}\) and \(z_{2}\), and the laser-electron interaction \(\mathbf{E}(t)\cdot\mathbf{r}\) and \(\mathbf{A}(t)\cdot\mathbf{p}\) are replaced with \(\mathbf{E}(t)z\) and \(\mathbf{A}(t)\mathbf{p}_{z}=-i\mathbf{A}(t)\partial/\partial z_{z}\), respectively, in Equations (28), (30) and (37). Orbitals are discretized on equidistant grid points with spacing \(\Delta z=0.4\) within a simulation box \(-1000\leq z\leq 1000\), with an absorbing boundary implemented by a mask function of \(\cos^{1/4}\) shape at 10% side edges of the box. Each EOM is solved by the fourth-order Runge-Kutta method with a fixed time step size (1/10,000 of an optical cycle). Spatial derivatives are evaluated by the eighth order finite difference method, and spatial integrations are performed by the trapezoidal rule. We consider a laser electric field given by  
\[\mathbf{E}(t)=\mathbf{E}_{0}\sin(\omega_{0}t)\sin^{2}\left(\pi\frac{t}{\tau}\right), \tag{43}\]  
for \(0\leq t\leq\tau\), and \(\mathbf{E}(t)=0\) otherwise, with a wavelength \(\lambda=2\pi/\omega_{0}=750\) nm, a foot-to-foot pulse length \(\tau\) of three optical cycles, and a peak intensity \(I_{0}=\mathbf{E}_{0}^{2}\) for \(I_{0}=5\times 10^{14}\) W/cm\({}^{2}\) and \(I_{0}=10^{15}\) W/cm\({}^{2}\). The 1D Hamiltonian, computational details, and the applied laser field are the same as used in Ref. [48] to facilitate comparison with TDSE results in Ref. [48]. First, we compare the time-dependent dipole moment \(\langle z\rangle(t)=\langle\Psi(t)|(z_{1}+z_{2})|\Psi(t)\rangle\) obtained with TDCIS approaches with that of TDSE in Figure 1, which immediately reveals a strong gauge dependence of fixed-orbital approaches, i.e., the large difference between LG and VG results. One should note that the comparison of LG and VG results alone can tell nothing about the preference of either approach; TDCIS method in both LG and VG are the first approximation in the hierarchy of CI expansions, which, at the full-CI limit, would be gauge invariant. The point here is that the LG scheme outperforms the VG scheme in comparison to the exact TDSE result as clearly seen in Figure 1, which convinces one of an _empirical_ preference of the LG treatment. On the other hand, the results of LG and rVG agree perfectly within the graphical resolution, numerically demonstrating the theoretical gauge invariance. Figure 1: Time evolution of the dipole moment of 1D-He exposed to a laser pulse with a wavelength of 750 nm and an intensity of (**a**) 5 \(\times\) 10\({}^{14}\) W/cm\({}^{2}\) and (**b**) 1 \(\times\) 10\({}^{15}\) W/cm\({}^{2}\). Comparison of the results with time-dependent configuration interaction singles (TDCIS) in the length gauge (LG), velocity gauge (VG), and rotated velocity-gauge (rVG) with that of the time-dependent Schrödinger equation (TDSE). Next, we consider the dipole acceleration \(\langle a\rangle(t)\) defined as the time derivative of the kinematic momentum,  
\[\langle a\rangle(t)=\frac{d\langle\pi\rangle}{dt}, \tag{44}\]  
where \(\langle\pi\rangle=\langle\Psi|(\mathbf{p}_{z_{1}}+\mathbf{p}_{z_{2}})|\Psi\rangle\) for the LG, and \(\langle\pi\rangle=\langle\Psi|(\mathbf{p}_{z_{1}}+\mathbf{p}_{z_{2}})|\Psi\rangle+2 \mathbf{A}(t)\) for the VG.