## 3 Solving the METDSE Nonperturbatively via the State-Specific Expansion Approach
### The State-Specific Expansion Approach (SSEA)  
For one-electron systems, the solution of Equation (1) for various problems has been achieved numerically, either via direct integration on a space-time grid or via a method of expansion over some type of one-electron basis functions. The grid method has also proven practical in certain cases involving closed-shell N-electron states, where, following the proposal and results of [PERSON] in the late 1980s [18; 19], the formalism is simplified by assuming that the field-driven integration involves only one active electron at a time (_single active electron_ (SAE) model). Space-time numerical integration over a grid in all dimensions of the partial differential Equation (1) has also been implemented in the case of the two-electron closed-shell \({}^{1}S\) state. (e.g., see [20], where differences between the SAE and the full two-electron results are attributed to electron correlations). The full grid approach is practically inapplicable to arbitrary N-electron electronic structures, i.e., beyond \(He\) or \(H_{2}\), even with the help of supercomputers. This limitation holds not only for the METOSE, but also for the eigenvalue problem of the METISE, as was recognized in the 1960s. Evidently, the theory for solving the METOSE for arbitrary electronic structures must follow a different path. This fact determined the research program we initiated in the early 1990s. We concluded that a robust and general approach would be one that invokes the fundamental expansion principle in terms of stationary states with complex time-dependent coefficients, and capitalizes on knowledge of, and experience with, calculating the stationary _state-specific_ N-electron wave functions of the discrete and of the continuous spectrum that solve the METISE [1; 2; 3; 21; 22]:  
\[ \begin{array}{l}{\bf H}\Psi_{k}(q)=E_{k}\Psi_{k}(q)\\ {\bf H}=\sum\limits_{i=1}^{N}{\mathbf{h}}_{i}+\sum\limits_{i>j}^{N}{\mathbf{g}}_{ij} \end{array}  \tag{2}\]  
\(k\) labels stationary states of either the discrete or the continuous spectrum. The latter are energy-normalized. \(q\) stands for the coordinates of the \(N\) electrons. In our SSEA work on the solution of Equation (1), the electron-field coupling operator in \({\mathbf{V}}(\omega,t)\) has been used either in the electric dipole approximation or as the full electric operator of the multipolar Hamiltonian [23] constrained by dipole selection rules ([24] and its references). The critical question is how to nonperturbatively solve the METOSE using \({\mathbf{H}}(t)\) and the many-electron \({\bf H}\) of Equations (1) and (2). For this purpose, we write the formal solution of Equation (1) as follows (here, for simplicity, I use only one-channel for the continuum):  
\[|\Psi(t)>=\sum\limits_{m}a_{m}(t)|m>+\int\limits_{0}b_{\varepsilon}(t)| \varepsilon>d\varepsilon \tag{3}\]  
where \(|m>\) are the stationary states of the discrete spectrum and \(|\varepsilon>\) are those of the continuous spectrum, including possible resonances. The scattering wave functions are energy-normalized according to \(<\varepsilon|\varepsilon^{\prime}>\) = \(\delta(\varepsilon-\varepsilon^{\prime})\). The complex coefficients represent the projection of the wavepacket \(|\Psi(t)>\) onto the corresponding stationary states. The objective of the SSEA is to represent the \(|m>\) and \(|\varepsilon>\) of Equation (3) by the state-specific wavefunctions that are relevant to the problem of interest, and to correctly compute the time-dependent coefficients. The latter task is accomplished by solving the coupled equations that result from substituting Equation (3) into the METDSE. The calculation is transparent for each trial set of state-specific wavefunctions, and is carried out by adjusting parameters such as the time step or the range of the free-electron energy and the number of points (solutions) in it, so as to achieve good convergence. This endeavor requires calculating energies and _bound-bound_, _bound-free_, and _free-free_ N-electron matrix elements [1; 2; 21; 22; 24].