## 3 Solving the METDSE Nonperturbatively via the State-Specific Expansion Approach
### Other Computational Schemes for Solving the METOSE for Problems of Atomic (Molecular) States  
_Interacting with Ultrashort and/or Strong Pulses_  
As was already stated in the previous sections, the aim of the present paper is to present elements of the theory and methods of the SSEA, and of quantitative results that the Athens team has produced in connection with problems that fall in the domain of astrophysics. Nevertheless, it may be of interest to the reader to cite other methods that have been proposed and used to solve the METOSE. Much of the text that follows is taken from [3]. As new experimental possibilities for the study of phenomena resulting from the interaction of atomic and molecular states with short and/or strong pulses were emerging, the introduction in 1994 of the SSEA [1; 2] was linked to the need at that time to develop theory and _many-electron_ methods of computation that are suitable for nonperturbatively solving the METOSE, beyond the cases of one- and two-electron ground states interacting with ultrashort and/or strong pulses. Regardless of the type of physical problem, in pursuing such a goal, the structure of the formalism must allow, in a practical way, the calculation of effects due to the _state-specific_ features of the pertinent electronic structures and electron correlations. This requirement is analogous to, but much more complex and demanding than, that of many-electron problems of quantum chemistry on the energy axis. Subsequent publications by a number of other groups have also presented methods aiming to provide solutions of the METOSE for many-electron systems. A recent review of a number of such methods was given by [PERSON] and [PERSON] [35]. Its subsections are titled \"Direct solution of the TDSE\", \"Multiconfiguration self-consistent-field (MCSCF) approach\", \"Time-dependent configuration interaction (TDCI) method\", \"Time-dependent multiconfiguration Hartree-Fock (TDMCHF) method\", \"Time-dependent complete active-space self-consistent field (TD-CASSCF) method\", \"Time-dependent coupled-cluster method\", and \"R-matrix approach\". In a note added in proof, the authors stated that their review did not cover the SSEA. I comment briefly on a few of these methods, starting with the _time-dependent R-matrix_ (TDRM) theory [36; 37; 38; 39]. The TDRM was first outlined by [PERSON] and [PERSON] in 1997 [36]. The fact that it was then applied only to a toy model, multiphoton ionization of a one-dimensional model, is testimony to the difficulties and high demands that such calculations have when it comes to problems involving real polyelectronic atoms or molecules. In a recent publication [39], the TDRM method was applied to a novel problem that is discussed in Section 6, namely the determination of the relative _time delay_ in the (2s,2p) photoionization of neon. Given the experience with advanced electron correlation methods for the ground and low-lying discrete stationary states and the convenience of available computer programs of computational quantum chemistry that are applicable to solving the METISE, in recent years many research groups have entered the area of solving the METOSE for certain problems by converting known time-independent methods into time-dependent ones in terms of expansions with time-dependent coefficients. For example, one is the TDMCHF (e.g., [40; 41; 42]), and another is the TDCI (e.g., [43; 44]). Even when the initial-state electronic structure is favorable for their formal implementation (i.e., single-determinantal closed-shell ground states), methods such as those in [40; 41; 42; 43; 44; 45] express the content of the SSEA only in an approximate way. However, as I already mentioned in Section 3.2,for the time being approximate treatments may indeed be necessary for the study of molecules.