@inproceedings{Nicolaides_2010, title={Theory and State-Specific Methods for the Analysis and Computation of Field-Free and Field-Induced Unstable States in Atoms and Molecules}, volume={60}, ISBN={978-0-12-380900-1}, ISSN={0065-3276}, archiveLocation={Ινστιτούτο Θεωρητικής και Φυσικής Χημείας (ΙΘΦΧ) - Επιστημονικό έργο}, url={https://hdl.handle.net/10442/12651}, DOI={10.1016/S0065-3276(10)60004-4}, abstractNote={The chapter discusses certain formal properties of field-free and field-induced unstable (nonstationary or resonance) states in atoms and molecules which are embedded in the continuous spectrum, and corresponding methods for the practical solution of the many-electron problem (MEP) that accompanies the requirement of the ab initio computation of their wavefunctions and of measurable quantities. It synthesizes new material and commentary with selected published theoretical and numerical results by the author and collaborators, going back to Nicolaides, Phys. Rev. A 6, (1972) 2078. Because there is a central theme that pervades the formalisms and applications of this work, its framework has been given the generic name of: The state-specific approach (SSA) (Nicolaides, Int. J. Quantum Chem. 60, (1996) 119; ibid, 71, (1999) 209). According to the SSA, critical to the development of formalism which is physically helpful as well as computationally practical is, first, the choice of appropriate for each problem forms of the trial wavefunctions and, second, the possibility of employing corresponding function spaces that are as specific and optimal as possible for the state and property of interest. A salient feature of the SSA is that it makes the interplay between electronic structure and dynamics transparent. At the core of the analysis and methods that are discussed in this Chapter is the consistent consideration of the fact that the form of each resonance wavefunction is Psi(r) = a Psi(0) +X(as) (Eq. (4.1) of text). If necessary, the extension to multi-dimensional forms is obvious. Depending on the formalism, the coefficient a and the “asymptotic” part, X(as), are functions of either the energy (real or complex) or the time. The many-body, square-integrable, Psi(0), represents the localized part of the decaying (unstable) state, i.e., the unstable wavepacket which is assumed to be prepared at t = 0. Its energy, E(0), is real and embedded inside the continuous spectrum. It is a minimum of the average value of the corresponding state-specific effective Hamiltonian that keeps all particles bound. The frameworks used in the SSA of this chapter are either energy-dependent or time-dependent, and engage methods that solve in a practical way the corresponding Schrodinger equations nonperturbatively. Both Hermitian and non-Hermitian formalisms have been applied, using the Coulomb or the relativistic Breit-Pauli Hamiltonians. As regards the theme of irreversible decay, the discussion examines briefly aspects of the relevance of three time domains: (1) The regime of exponential decay (ED), e(-Gamma t), which is the regime that defines the resonance state on the energy axis in terms of its energy, E(r), and its width, Gamma. (2) The regime of pre-ED, which defines the degree of stability of a(t)Psi(0) subject to interactions with X(as)(t), and (3) The regime of non-ED that starts after a duration of (normally) many lifetimes in the ED regime. The information on the bulk of the interparticle interactions is contained in Psi(0), even though the decay properties are obtained only after input from X(as) has been added. It follows that its accurate and systematic computation is of prime importance. Instead of following procedures of brute-force diagonalization, the calculation of Psi(0) for electronic structures is done directly, via the solution of state-specific Hartree-Fock (HF) equations under special orbital constraints. The practicality of this approach even for open-(sub)shell multiply excited resonances was demonstrated in the 1972 paper, thereby opening the way for later computations that start with state-specific multiconfigurational HF equations and tackle efficiently the MEP and electron correlation in the context of appropriate formalisms for resonance states. When a Hermitian, real energy formalism is adopted, the mixing of bound-scattering components in the case of field-free resonances is computed from a multichannel (in general) K-matrix formalism, with frozen-core, term-dependent, energy-normalized HF scattering orbitals as basis sets for the continuous spectrum. On the other hand, following our work from the 1970s and early 1980s, it is shown that starting from Fano’s 1961 Hermitian formalism which accounts rigorously for discrete-continuum mixing, when the appropriate boundary conditions are imposed formally on the solution of the Schrodinger equation in the vicinity of E(0), two adjoint complex eigenvalue Schrodinger equations (CESE) emerge naturally, whose eigenfunctions have a two-component form, while their eigenvalues are complex conjugate. The physically acceptable solution for a decaying state, Psi(r), is the one with the outgoing-wave boundary condition, for which the complex energy is z(r) = E(r) - (i/2)Gamma. Methods have been developed and applied for the non-perturbative solution of the corresponding state-specific CESE for both the field-free and the field-induced cases, within non-Hermitan formalisms with optimized superpositions of real and complex functions of real or complex coordinates. In summary, the discussion includes formalism and analysis contributing to the understanding of the nature of unstable states, as well as indicative theoretical and numerical examples from applications to atoms and molecules, and related comparisons with other methods, concerning prototypical problems of autoionization, predissociation, series of isolated and overlapping resonances, structure and spectroscopy of doubly and multiply excited states, multiphoton ionization, field-induced polarization, etc.}, booktitle={Advances in Quantum Chemistry, Vol 60: Unstable States in the Continuous Spectra, Part 1: Analysis, Concepts, Methods, and Results}, publisher={Elsevier Academic Press Incorporated}, author={Nicolaides, Cleanthes A.}, editor={Nicolaides, CA and Brandas, EEditors}, year={2010}, pages={163–267} }