## 5 Time-Resolved Coherent Excitation and Decay of Strongly Correlated Autoionizing States  
Over many decades, information on properties of resonance (autoionizing) states inside the continuous electronic spectra of real N-electron atoms and molecules has been obtained, to various degrees of accuracy, on the _energy axis_ (e.g., energy positions, partial and total energy widths, photoionization cross-section in the energy region where a resonance is present, perturbation by an external field). This area of research has seen significant experimental and theoretical progress. Regarding theory, the fundamentals of resonance-state theory are settled. However, when it comes to practical implementation for problems involving complex excited electronic structures and multichannel continua, what is needed in addition to the fundamentals is the possibility of reliable calculations. Indeed, this direction contains many possibilities for application to complex situations. In the mid-1990s, a new direction in the investigation of N-electron autoionizing (resonance) states was initiated. It was based on the many-electron calculation of the N-electron \(\Psi(t)\) via the SSEA on the _time axis_[12; 13], a possibility that provides a different type of insight into their properties. For the treatment of autoionization, the Hamiltonian is field-free and time-independent, \(\mathbf{H}\). The initial state is an N-electron wavepacket, \(\Psi_{0}\), which is square-integrable at \(t=0\) and has energy \(E_{0}=\mathbf{\
abla}_{0}|\mathbf{H}|\Psi_{0}>\). The main goal in [12; 13] was to calculate the time-resolved _survival probability_ against electron emission, \(\left|<\Psi_{0}|\Psi(t)>\right|^{2}\), from which information such as the _lifetime_, or the _energy width_ corresponding to exponential decay of the decaying state, can also be obtained. The temporal resolution with which \(\left|<\Psi_{0}|\Psi(t)>\right|^{2}\) is obtained can be refined down to the attosecond scale. The prototypical cases that were studied fall into two categories of processes of dissipation into the electronic continuum:  
(1) _One-electron tunneling_ from a core-excited _shape_ resonance, \(He^{-}1s2p^{2}\,{}^{4}P\), according to \(He^{-}1s2p^{2}\,{}^{4}P\)\(\xrightarrow{\text{auto}\,\text{det. }\,\text{achment}}\)\(\left[He\,\,1s2p\,\,{}^{3}P^{o}+\varepsilon p\right]\,{}^{4}P\)[12]. This decay process is due mainly to the state-specific self-consistent field changing from initial to final state, and can be (was) calculated as such. (2) _Two-electron rearrangement_. This is the \"mechanism\" that normally dominates nonrelativistic or relativistic autoionization. For example, an ab initio calculation produced the time-resolved decay curve and the lifetime, \(3.5\times 10^{-14}\) s, of the doubly excited state \(Ca\,\,KL3p^{6}3d5p\,\,{}^{3}F^{o}\)[13]. The results provided, for the first time, quantitative results for real N-electron systems regarding the time-resolved regimes of both the exponential decay and, most importantly, the nonexponential decay (NED) [12; 13]. I note that the states were not chosen haphazardly. Instead, their choice was based on the theoretical results and proposal published in 1977 [48], according to which, for NED of an isolated decaying state to acquire a magnitude of possible physical significance, the energy, E, of the state must be very close to threshold and the ratio \(\mathrm{R}=\mathrm{E}/\Gamma\) must be close to unity. (\(\Gamma\) is the width of the isolated narrow resonance state.) For a recent application of the results of [48] to radioactivity and carbon dating, see [49]. The insight and computational experience gained from the ab initio calculation of the time-dependent wave functions of autoionizing states, including the effects of electron correlation and of their time-resolved decay [12; 13], served as input for the theoretical and computational work we initiated immediately after the important publications by [PERSON] et al. [50] and [PERSON] et al.