## 3 Solving the METDSE Nonperturbatively via the State-Specific Expansion Approach
### The State-Specific Expansion Approach (SSEA)
One of the advantages of constructing expansion (3) in terms of wavefunctions that are calculated separately for each state, in terms of their own zero-order and correlation function spaces, is that in the process of solving, one can monitor and evaluate with transparency and economy the dependence of the evolution of \(\Psi(t)\) on each \(|m>\) and \(|\varepsilon>\). In other words, unlike other methods that are often used for the approximate solution of Equation (1), the SSEA is not a \"one-shot\" algorithm that is applied in the same way to different cases. Instead, the methodology involves systematic expansion in terms of state-specific N-electron wavefunctions for the discrete and the continuous spectrum, according to the physics of each problem and the observed convergence of the calculation. Furthermore, the SSEA is not limited to special electronic structures that are described by a single determinant, nor does it ignore the variety of effects of electron correlations, multiply excited resonances, and interchannel couplings. Once the expansion coefficients of Equation (3) have been obtained from the solution of Equation (1), it is straightforward to use them, together with the stationary wavefunctions, in order to unambiguously compute the dynamics of interest as a function of time during and at the end of the pulsed interaction. This simple and direct way of quantitatively recognizing the correspondence of the stationary states with the field-induced dynamics is a hallmark of the SSEA. Its utility has been demonstrated in a variety of applications to difficult and computationally demanding TDMEPs, where, among other things, insight and experimentally relevant quantitative information has been gained by using the SSEA coefficients, as, for example, in cases of photoelectron angular distributions, above threshold ionization, the contribution of Rydberg states in multiphoton ionization, and the relative time delay in the emission of photoelectrons from different subshells (see [1; 2; 3; 4; 5; 6] and Section 6 of this paper). The solution of the METDSE according to the SSEA takes into account the following:  
* The zero-order features of electronic structures of initial, intermediate, and final states. These can be calculated at the Hartree-Fock (HF) or multiconfigurational HF (MCHF) level (see Equation (4) below). * The dominant electron correlations for those states (initial, intermediate, or final) where this is necessary. * The contribution of perturbed or unperturbed Rydberg levels, and of doubly or even multiply and inner-hole excited states. * The contribution of the continuous spectrum of energy-normalized, channel-dependent scattering states, without or with resonance states. * The interchannel coupling. It is important to add that the structure of the SSEA allows the use of relativistic wavefunctions as well. Molecules  
As regards molecules, the SSEA has been implemented only for the purpose of treating special themes involving vibrational states and dissociation continua in diatomics, where the smooth continuum has been described in terms of either box-normalized basis sets [1; 25] or numerical energy-normalized scattering states [26]. It has not yet been applied to problems of time-resolved electron dynamics in molecules, for which descriptive discussions can be found (e.g., in [27]). Given our limited human resources, our prospects of tackling such problems in molecules at the level of rigor that has been demonstrated for atoms via the SSEA are meager. However, the following comment is relevant. Tracking the hyperfast motions of electrons in molecules while discerning their effects from those caused by the other degrees of freedom is not easily amenable to rigorous computational analysis by accurately solving the METDSE, especially when the molecule is excited by a pulse of relatively short wavelength to a multitude of interacting states and channels. Of course, approximate treatments can, in principle, produce rough answers.