## 3 Solving the METDSE Nonperturbatively via the State-Specific Expansion Approach  
### Introduction  
The fundamental equation that contains information about the properties and phenomena that are observable when a pulse of radiation interacts with an atomic or molecular state is the METDSE:  
\[\begin{array}{l}\mathbf{H}(t)\Psi(t)=i\hbar\frac{\partial\Psi(t)}{\partial t}\\ \mathbf{H}(t)=\mathbf{H}+\mathbf{V}(\omega,t)\end{array} \tag{1}\]  
\(\mathbf{H}\) is the N-electron field-free Hamiltonian, consisting of one- and two-electron nonrelativistic or relativistic operators. \(\mathbf{V}(\omega,t)\) is the interaction one-electron operator. Apart from time and the photon frequency \(\omega\), it contains information about the polarization and intensity of the field, and about the temporal characteristics of the pulse.  
The possibility of solving the METDSE for TUMEPs in real systems lies at the heart of many-electron quantum mechanics. For many decades this goal was ignored, not only because of the complexity of the TDMEP, but also because the understanding of properties and phenomena did not require quantitative knowledge of the many-electron \(\Psi(t)\) of Equation (1). Instead, the focus was on the possibility of obtaining accurate solutions of the _many-electron time-independent Schrodinger equation_ (METISE) for the calculation of properties of stationary states, including time-independent rates of transition, an activity that remains dominant, especially for N-electron ground states.  
Given the countless number of possible many-electron states and spectra and the variety of possible radiation pulses, a generally applicable theory for the ab initio solution of Equation (1) must be able to incorporate in a tractable way the important requirements of advanced theory of electronic structures, of electron correlations, and of the scattering continuum.  
It should be recalled that most publications and books related to this subject ignore the serious and challenging issue of properly solving the TDMEP. In other words, no N-electron wave functions are calculated and used. Instead, these publications are exhausted at the level of either pure formalism or of phenomenology. In such categories are many discussions based on two- or three- or four-level models, where the N-electron matrix elements and energies are taken to be parameters.  
In other types of investigations, the complexity of the many-electron problem is bypassed by conveniently replacing the true electronic Hamiltonian with a solvable _independent electron model_ (IEM). IEM-type methods of solving Equation (1) are computationally convenient when applied to closed-shell (single determinantal) electronic structures. (When it comes to open-shell initial states, in most cases electron couplings are such that the zero-order description in terms of a single determinant is inadequate.) They may yield gross information for certain observables, analogous to the IEM that is used for the approximate solution of the METISE for closed-shell ground states. However, they cannot handle questions regarding the effects of electron correlations, of double (or multiple) excitations, and of interchannel couplings in the discrete and the continuous spectrum. Hence, unless one has a good understanding from experience or formal analysis of the situations for which the IEM may be reasonably accurate and meaningful, predictions for arbitrary systems based on IEM-type approaches are either practically impossible or unreliable.