# Gauge-Invariant Formulation of Time-Dependent Configuration Interaction Singles Method
## 1 Introduction  
Time-dependent configuration interaction singles (TDCIS) method is one of the powerful ab initio methods to investigate laser-driven electron dynamics in atoms and molecules [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. In the TDCIS method, the time-dependent electronic wavefunction is given by the configuration interaction (CI) expansion,  
\[\Psi(t)=\Phi C_{0}(t)+\sum_{i}^{\rm occ}\sum_{a}^{\rm irr}\Phi_{ia}C_{ia}(t), \tag{1}\]  
where \(\Phi\) is the ground-state Hartree-Fock (HF) wavefunction, and \(\Phi_{ia}\) is a singly-excited configuration-state function (CSF), replacing an occupied HF orbital \(\phi_{i}\) in \(\Phi\) with a virtual (unoccupied in \(\Phi\)) orbital \(\phi_{a}\), and the electron dynamics is described through the time evolution of the CI coefficients, \(C_{0}\) and \(\{C_{ia}\}\). The virtual orbitals \(\{\phi_{a}\}\) consist, in theory, of an infinite number of bound and continuum orbitals. In practice, one needs to work within a given, finite number of basis functions or numerical grids to represent orbitals; however, the real-space implementation with an appropriate absorbing boundary has been proved to correctly model both bound and continuum states, allowing to describe electron dynamics involving both (single) excitations and ionizations [4]. Compared to more involved ab initio wavefunction-based approaches [25] such as time-dependentmulticonfiguration self-consistent-field (TD-MCSCF) methods [26; 27; 28; 29; 30; 31; 32; 33], time-dependent \(R\)-matrix based approaches [34; 35; 36], or time-dependent reduced density-matrix approach [37; 38], distinct advantages of the TDCIS method include a low computational cost and the conceptual simplicity to analyze simulation results. Furthermore, an effective one-electron theory with coupled channels has been developed [2], which introduces the orbital-like quantity, called channel orbital,  
\[\chi_{i}(\mathbf{r},t)=\sum_{a}\phi_{a}(\mathbf{r})C_{ia}(t), \tag{2}\]  
and equivalently rewrites EOMs for CI coefficients with those for channel orbitals \(\{\chi_{i}(\mathbf{r},t)\}\) with no individual reference to virtual orbitals. This reformulation removes the bottleneck of the CI coefficient-based TDCIS method to compute all (or, at least sufficiently many, including bound and continuum) virtual orbitals prior to the simulation, and thus particularly useful in grid-based simulations. Despite this advantage, numerical applications of the channel orbital-based TDCIS method has been limited to Ref. [2; 14; 15] for a one-dimensional Hamiltonian and Ref. [1] for noble gas atoms with a Hartree-Slater potential, as far as we know, and the vast majority of applications to date have adopted the CI coefficient-based approach [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24], except for the use of \(\{\chi_{i}\}\) as intermediate quantities in evaluating photoelectron spectra [18]. The preference of CI coefficient-based approach might be partially due to the high symmetry of atomic systems, for which the stationary Hartree-Fock operator decouples for different angular momenta [4], making it a relatively feasible task to obtain all virtual orbitals (within a given radial grids or radial basis functions) for the lowest few angular momenta. The channel orbital-based approach would be more suited, on the other hand, to simulations of electron dynamics with intense and/or long-wavelength laser fields, requiring much longer angular momentum expansion [39; 40; 41], and moreover to grid-based molecular applications, where obtaining a sufficient spectrum of virtual levels could be unacceptably expensive. However, the TDCIS method, whether in the CI coefficient-based or channel orbital-based formulation, suffers from the lack of gauge invariance, as a general consequence of relying on truncated CI expansion with time-independent orbitals, or fixed orbitals. Previously, the length gauge (LG) has been employed, e.g., in Ref.