Quantum chemistry and its “ages”


  • This article was published online on 24 February 2014. An error was subsequently identified. This notice is included in the online and print versions to indicate that both have been corrected on 12 March 2014.


The popular methods of computational quantum chemistry (CQC) have acquired the status of “mainstream” quantum chemistry (QC), with countless useful applications to molecular science, especially for properties, potential energy surfaces, and reactions of the ground states. In earlier publications, four “ages of QC” have been defined, the classification having been based exclusively on the progress of QC in the direction of CQC.

The core of the present article has a dual character: On the one hand, it is a commentary on the nature of QC, whereby it is argued that modern QC, while keeping its focus on the many-electron problem, (MEP), and on the consequences of Electron Correlations on observable quantities, has a domain and a scope that are larger than those determined, more or less, by CQC. Hence, additional “ages” are singled out and proposed, which are connected to “time-independent” and “time-dependent” theoretical “many-electron” formulations and calculations in which the “continuous spectrum” is involved explicitly. They are connected to experimental directions, which challenge theory to provide not only phenomenology but also quantitative answers and predictions for properties and phenomena involving various types of “unstable (nonstationary) states,” which decay irreversibly into the continuous spectrum either via interactions within the atomic (molecular) Hamiltonian (e.g., autoionizing states) or via the interaction with external electromagnetic fields.

On the other hand, the article presents, in the form of brief reviews and retrospective accounts, a gleaning from our contributions to the theory and to methods of calculation for the quantitative treatment of such MEPs. Specifically, in the context of the previous paragraphs, I comment briefly on basic concepts, I point to early theoretical work, and, in support of the arguments, I refer to practical theoretical constructions and sample results regarding the following two themes: “resonances in many-electron systems” and “theoretical time-resolved many-electron physics.”

Both the themes are connected to the recent developments on the experimental front of the interaction of atoms and molecules with ultrashort radiation pulses (weak or strong) in the femtosecond and attosecond regimes. The first theme is treatable within energy-dependent or time-dependent frameworks. The second one requires the computation of a physically transparent Ψ(t), which solves to a very good approximation the many-electron time-dependent Schrödinger equation (METDSE) for given characteristics of the pulses.

Our approaches to related problems have been developed and implemented within state- and property-specific formulations. These are constructed in terms of energy-dependent Hermitian and non-Hermitian formalisms, as well as in terms of time-dependent ones, whose essential elements are explained here. In either case, in line with the main tenet of QC, the focus is on the understanding and efficient solution of the corresponding MEPs, especially in terms of nonperturbative methods. For time-dependent problems involving the interaction with strong fields, the solution of the relevant METDSE is carried out by applying the “state-specific expansion approach.” This type of calculation entails the solution of many thousands of coupled equations, where the input consists of bound–bound, bound–free, and free–free coupling matrix elements, on- and off-resonance with respect to the frequency of the pulse.

As an example of a solution of a time-resolved many-electron process, I present snapshots of the time-dependent formation of the asymmetric profile of the He 2s2p 1P0 resonance state excited by a femtosecond pulse. This profile is formed within about 180 fs and is the same as the one which is well-known from theory and experiment on the energy axis.

The domain and horizon of modern Quantum Chemistry are broader than those of its original component, namely, Computational Quantum Chemistry of the ground state. Hamiltonians can create energy- or time-dependent unstable states whose effects are measured inside the continuous spectrum. The understanding and practical solution of the corresponding many-electron problems can be carried out reliably, using state-specific, nonperturbative methodologies that solve the Schrödinger equation efficiently as a function of real or complex energies, or of time. © 2014 Wiley Periodicals, Inc.

On the Domain and “Ages” of Quantum Chemistry

Heitler-London's study of the hydrogen molecule in 1927 opened the new field of quantum chemistry, which is still under rapid development.

Quantum chemistry deals particularly with the electronic structure of atoms, molecules, and condensed matter, and describes it in terms of electronic wave patterns of standing waves. It deals also with collisions between atoms and molecules and with the study of chemical reactivity.

P. O. Löwdin, 1991[1]

In the following paragraphs, I take advantage of the accommodation provided by the flexibility of conference proceedings, (Eighth congress of the International Society of Theoretical Chemical Physics, Budapest, August 25–31, 2013), to indulge in a brief commentary on the nature and domain of the broad discipline with the established name “quantum chemistry” (QC).

The title of the article was inspired by the title of a 2012 review article of Császár et al.,[2] which discusses the theory and methods for solving the time-independent Schrödinger equation (TISE) with the Hamiltonian including terms that account for effects of “nuclear motions.” This permits the ab initio calculation of accurate ro-vibrational spectra, once the corresponding accurate potential energy surfaces are available. This type of research was classified as the “fourth age of QC,”[2] in continuation of the title of a brief commentary by Richards[3] in 1979, who pointed to the then emerging evidence, based on the results of accurate calculations by Wetmore and Schaefer on acetylene, that a “third age of QC” had dawned in the mid-1970s, where, “calculations are more accurate than experiment or at the very least sufficiently accurate to be indispensable for interpretation.” In stating such a characterization, Richards essentially identified, in agreement with established lore, the course of the development and achievements of QC exclusively with the theory and methods that had tackled, at levels of accuracy progressively higher, the problem of computing energies and other properties of molecular structures in the ground state and, occasionally, in a couple of low-lying excited states, for example, [ [1-8].

The nature of problems and the types of methods that are typical in the works, which are discussed and cited in [ [1-8] are, by now, recognized as the core of “computational quantum chemistry,” (CQC), which has been dominating QC for decades, in terms of number of practitioners, of publications and of quantity of information. It has essentially become “mainstream QC,” which grew out of the theoretical formulations that were created and demonstrated mainly during the 1960s–1970s, (Nobel prize in Chemistry[5, 6]). Thus, it may seem natural to many researchers to associate the “ages” of QC with those of CQC and to choose the themes with which QC “deals” as in the Löwdin quotation above.

In contemporary times, the methods of the conventional CQC fall into two major categories: one category consists of methods aiming at the solution of the TISE via the calculation of hierarchical wavefunction (energy) expansions containing “virtual excitations” of progressively higher order, based on a zero-order reference wavefunction that is supposed to be computable with accuracy. The other uses various adjustable functionals in the framework of “density functional theory” (DFT).[4, 6, 8] In either case, the “Holy Grail” is the possibility of computing as well as possible electron correlation energy (ECE), a phrase that is often used as the alternative to the phrase “approximate solution of the many-electron problem” (MEP).

The original definition of ECE and corresponding formulations of many-electron methods start from the Hartree–Fock (HF) solutions of closed-shell ground states, such as those of the 10-electron Ne, H2O, or CH4. Parallel to this approach, the methods of DFT are also in the business of dealing with the ECE implicitly, in terms of density functionals that are often adopted according to empirical data. The closed-shell character of the ground states of a large number of molecules facilitates formalism and computation, as the (restricted) HF zero-order wavefunction is well defined and its calculation is, by now, possible even for molecular systems with very many electrons. This fact has allowed the production of many quantitative results, with associated systematic studies with respect to different basis functions and to “chemical models.”[5]

Conversely, other classes of states involve, as a function of the parameters of the Hamiltonian such as nuclear charge and geometry, various types of wavefunction mixings in zero order, and so their overall treatment must be more sophisticated. This direction of QC, where analysis, formalism, and computation use a multiconfigurational zero-order wavefunction that are expected to carry the dominant physicochemical information as a function of geometry, has already been proven useful and expedient. For a recent discussion of this aspect of the theory and calculation of electronic structures, with illustrations of the concepts of “nondynamical” and “dynamical” electron correlation (EC), and their understanding with respect to the concepts of “Fermi-sea” and of “complete active space,” the reader is referred to the recent review of the “state- and property-specific Quantum Chemistry” (SPSQC).[9] A salient feature of the SPSQC is that it makes (or aims at making) the computation of the interplay between various electronic structures and dynamics tractable, economic, and physically transparent. The few examples from our work, which are cited and/or discussed in the following sections are results of the SPSQC.[9-11]

My claim here is that, although QC obviously includes CQC, it also goes beyond its formalisms, methods of computation, and practical goals, especially as experimental developments keep opening new frontiers. This does not gainsay the enumeration of “ages” of QC as in [ [2, 3]. However, it does render it precarious from the point of view that, as I state above and elaborate below, although CQC has indeed become “mainstream QC,” it omits other significant and challenging avenues of research that have been contributing their own “ages” to modern QC.

Vistas of QC: Solving the MEP for situations defined by new experimental techniques of excitation and of measurement on the energy as well as on the time axes

It is reasonable to argue that the “trade mark” of QC is the pursuit of reliable solutions of the MEP, as it manifests in various types of properties and phenomena, in small or large systems, without or with the interaction with external weak or strong electromagnetic fields. In the standard Born–Oppenheimer framework, accurate calculation of effects of nuclear motion presupposes the use of reliable solutions of the MEP.

In general, the incessant achievements and motion into new directions of experimental and theoretical basic science generate new frontiers and possible new “ages” from time to time, albeit with different rates of growth and different numbers of people engaged in related research. At the same time, partial overlaps of domains and use of similar methods between differently named disciplines take place so that permanent assignments and boundaries cannot be sustained. For example, a problem of solid state physics may sometimes also be tackled by methods and analyses more akin to ground-state QC (e.g., use of localized orbitals or models of big molecules and clusters). Of course, there are certain hallmarks that can be accepted as providing the measure and the general perimeter. For example, nuclear physics is distinct from atomic physics, even though the study of hyperfine structure may bring people from both the fields to a common ground. Similarly, it makes sense to choose the two fundamental equations of Schrödinger, without or with relativistic formulations and adjustments due to Dirac, Breit, and Pauli, as those which characterize QC as well as much of theoretical atomic, molecular, optical, and chemical (AMOC) physics. In other words, the criteria of delineation may become blurred or deficient when named disciplines, such as QC, are meant to be associated only with specific systems and only with preassigned issues and problems.

As regards QC, limiting its nature, its domain, and its “ages” to theories and many-electron, many-nuclei, time-independent computations involving essentially only ground states (and reactions between them) or, at best, low-energy discrete excitations and related spectroscopy, does not tell the full story and does not give the whole picture. Even going back to the “origins,” a more representative and more challenging QC can find its “ancestral roots” not only in the pivotal Heitler–London paper on the quantum explanation and semiquantitative description of the covalent bond of H2, to which Löwdin[1] as well as other authors refer to set the stage of the topics with which QC “deals” but also in other early papers whose contents and objectives were then (late 1920s–early 1930s) quite far apart and disparate but are not so today.

In support of the above statement, I offer the following example from the early period of quantum mechanics, which can be linked to modern QC: on one hand, fundamental questions of the computation of electronic structures, of open-shell excited states and of EC were dealt in early papers of Heisenberg, Slater, Hartree, Fock, Hylleraas, and others. On the other hand, during the same period, the concept and theory of two-photon transitions was introduced by Göppert-Mayer, long before such and higher order processes were measured and studied computationally in many-electron atoms and molecules. In our times of the computer, of the laser, and of the large variety of actual and possible experiments on effects of matter–radiation interaction, the need to understand quantitatively, from a many-body point of view, and within time-independent as well as time-dependent frameworks, multiphoton processes in multielectron atoms and molecules is very real. This is certainly within the domain of a modern QC that goes far beyond the calculation of, say, ECE and electronic structure of ground states. In other words, if one wishes to explain the nature, domain, and scope of QC in terms of a historical retrospection to certain early quantum mechanical publications that may be considered as the critical origins of its path, the Heitler–London paper is obviously fundamental but is not the only one.

Continuing along these lines, let me point to the standard issues of bond formation and of bond cleavage, which is the prime example of “down to earth” chemistry. It is obvious that the introduction and continuing evolution of “laser spectroscopy” of all kinds (e.g., in terms of available radiation wavelengths, pulse durations, and intensities) has revolutionized the possibilities of study of new electronic structures and their properties, of new types of atomic and molecular spectra, and, therefore, of new channels for excitation and evolution, and for bond fragmentation and formation (e.g., areas of photophysics and photochemistry). It is then reasonable to argue that such developments, whose foundations are to be found in the explosion of methods of “excitation” and of measurement with high resolution on the energy as well as on the time axes, ought to play a role in defining (without sharp boundaries) the domain of modern QC and in identifying its “ages.”

For example, in the 1960s, the introduction of spectroscopy using as probe synchrotron radiation opened the horizons for the systematic study, over a wide range of the energy spectrum, of spectra associated with a variety of multiply excited or of inner-hole excited states lying in the continuous spectra of atoms and molecules and their ions. Similarly, beam-foil spectroscopy became a most effective and productive tool for the measurement of radiative lifetimes of all types of excited states, in neutral atoms and molecules, as well as in negative ions, and in a huge range of positive atomic ions.

In recent times, spectacular advances have been achieved in the science and technology of production and use of new sources of radiation pulses (e.g., free-electron laser) whose intensity can be very strong, of ultrashort duration, and of wavelengths covering a huge portion of the electromagnetic spectrum. Corresponding pump-probe techniques of measurement can yield novel information for various elementary processes that can be time-resolved even on the attosecond scale (1 as = 10−18 s). Thorough reviews of these new experimental directions, covering a spectrum of fascinating and challenging topics, have recently been published by Krausz and Ivanov[12] and by Agostini and DiMauro.[13]

The MEP in field-free and field-induced unstable states in the continuous spectra. Time-resolved phenomena in terms of nonperturbative solutions of the METDSE

Considering just the examples given above, it becomes clear that the application of appropriate experimental arrangements using a variety of probes that have been made available over a period of a few decades since the 1960s (including highly monochromatic electron beams, whose use can uncover resonances in negative ions) have opened the door for the creation, observation, and control of a multitude of quantum channels of state-excitation, evolution, reaction, and eventual relaxation into continuous energy spectra.

Now, new types of quantum states necessarily become candidates for theoretical and computational studies in frameworks that not only treat the phenomenology but also, and this is where the forte of quantum chemists is supposed to play its role, tackle efficiently the corresponding MEPs. If viewed from a time-dependent viewpoint, where the initial state is a localized wavefunction (many-electron in general) their eventual decay into the continuous spectrum, places them in the category of field-free or field-induced “unstable states in the continuous spectra” (USCS). This multifaceted major category of AMOC physics was recently reviewed in articles by a number of experts.[14, 15]

Understanding quantitatively the properties of these states and their role and contribution to various phenomena, requires the development and efficacious application of new formal and computational aspects of time-independent many-electron quantum mechanics as well as of explicitly time-dependent many-electron quantum mechanics, that is, of broad areas of research, which differ substantially and are more complex than “mainstream QC,” whose only concern from the point of view of fundamental theory is the convergence of one energy solution of the TISE to the (in principle unknown) numerically exact eigenvalue.

Therefore, I opine that additional “ages of QC” can be recognized, provided they are characterized by their focus on the solution of the omnipresent MEP, which is the traditional backbone and signature of QC. The most fundamental of such problems is the possibility of solving, formally and computationally, the many-electron time-dependent Schrödinger equation (METDSE) to all orders in perturbation theory, or, equivalently, nonperturbatively. Needless to add, starting from the METDSE reductions to time-independent formulations are in order when appropriate.

Following is a list of loosely categorized topics, which, according to the previous arguments, require frameworks of QC, which are different than that of CQC:

  1. Highly excited states near the fragmentation threshold. These appear in, for example, perturbed multichannel Rydberg-like spectra, in questions of formation or not of stable “negative ions” in ground or in excited states, in ro-vibrational spectra of electronically highly excited molecules, in dynamics near the maxima of field-free or field-induced multidimensional (in general) potential barriers and so forth.
  2. Unstable (nonstationary) states in the multichannel (in general) continuous spectrum. In the context of different theoretical constructions and experimental measurements, these are normally called “resonance,” or “autoionizing,” or “quasi-discrete,” or “Auger,” or “compound,” or “predissociating,” or “collision complex” states. A distinct category is that where the instability and dissolution into the continuous spectrum is caused by the interaction with external electromagnetic fields. Time-independent (Hermitian and non-Hermitian), as well as time-dependent frameworks are available.
  3. Perturbations of ground or excited states by strong (relative to the type of initial state), electromagnetic fields. Time-independent as well as time-dependent frameworks have been published and tested, accounting for the interaction either to all orders (variationally) or to very high orders of perturbation theory.
  4. Solution of the METDSE, where the Hamiltonian may be nonrelativistic or relativistic, and possibility of time resolution of the effects of strong EC s at femtosecond and, especially, attosecond time-scales. Ab initio and mathematically correct incorporation of the contribution of the multichannel (in general) continuous spectrum.

Having offered the preceding arguments, in the following sections, I turn to a quick gleaning from the many topics discussed recently in [9-11] within time-independent (energy-dependent) and time-dependent frameworks of the SPSQC. My commentary is relatively brief, and relates to the above topics (1–4), with emphasis on the state-specific approach to the MEP for resonances in multielectron atoms and molecules and on elements from the contents of topic (4).

The review articles[9-11] contain old as well as new material, many references, contextual analyses, numerical applications to prototypical systems, comparisons of our results with those from other methods and experiments, and synopses of the relevant historical background.

Many-Electron States Near the Fragmentation Threshold Perturbed by the Continuous Spectrum

As I have already indicated, the focus of my discussion as regards the treatment of MEPs is not on the QC of ground states of neutrals with large number of electrons. Rather, it is on the QC that is needed for the solution of time-independent or time-dependent problems, where excitation takes place and field-free or field-induced unstable states are created in the continuous spectrum. I note that inside a multichannel continuum, there are also states that are bound (they belong to the discrete spectrum of the Hamiltonian) due to symmetry restrictions. Both the types normally have open-shell electronic structures with possibly strong ECs. In this context, few-electron systems usually offer opportunities for testing and illustrating concepts, theories, and methods of QC, where applicable.

I will comment briefly on aspects of results on states of negative ions, either as ground or excited bound states just below the corresponding threshold or as resonance states. Given the nature of the article, only a few relevant references are cited. The rest can be found, together with additional information on the theme of resonances in negative ions, in the reviews [9, 10].

Negative ions of atoms and molecules constitute a class of systems with special spectral and physicochemical properties. The requirements for reliable theoretical and computational approaches are, in general, more demanding than those applied for the treatment of ground states of neutrals, regardless of the overall number of electrons. Three reasons why this is so, either for bound states or for transient resonances of negative ions, are the following:

  1. The consequences of ECs are usually more drastic than those of the corresponding neutral species, as the effects of interelectronic couplings and interactions for the added electron(s) are now prevalent. As a consequence, one should expect that the radial parts of one or two outer orbitals will lose, to a degree that may complicate the calculations, their distinct features that are known from the electronic structures of neutral atoms and small molecules. For example, outer orbitals of resonance states of atomic negative ions (ANIs) need not have the exact radial features of hydrogenic shell structure.
  2. Regardless of the degree of accuracy of an energy calculation, to secure the reliability of calculations of properties that depend on the details of wavefunctions, the methodology must additionally achieve the numerically accurate calculation of the asymptotic part, a condition that is normally of lesser significance in the usual calculations of ground states of neutral atoms and molecules.
  3. The continuous spectrum normally acquires significance that must be dealt with, formally as well as computationally, even for those states that turn out to be just below their ground state threshold, that is, they are stable (e.g., see the discussion in the subsection below). Needless to add that, when it comes to the resonance states one cannot ignore the contribution of the continuous spectrum, which must be incorporated in a mathematically correct way.

A practical and general solution to problems associated with items (1) and (2) above is to calculate the zero-order wavefunction (single- or multiconfigurational) self-consistently and numerically, as it is done in the SPSQC for atoms and diatomics.[9-11] For example, I can refer to the solution of two totally different problems: See [ [16] for state-specific calculations aiming at determining the electronic structures and the existence or not of certain resonance states in the diatomic negative ion math formula. Also, see [17] for the understanding of problems concerning the accuracy of multiphoton rates of detachment in the prototypical H and their solution in terms of appropriate many-electron, many-photon theory.

The understanding and solution of problems associated with all items (1, 2, and 3) requires the combination of fundamental aspects of scattering theory with those of polyelectronic theory of electronic structures. It is with this requirement in mind that the SPSQC theories and computational methodologies reviewed in [ [9-11] were created and developed, with numerical applications to prototypical systems.

In the following examples, the emphasis is on the solution of MEPs, rather than on phenomenology based on models. The structure of the theory and methods is based on the calculation of matrix elements between state-specific wavefunctions. The coupling operators may be either nonrelativistic or relativistic. In addition, for diatomics, where the decay of unstable states into the continuous spectrum may also occur via ro-vibrational coupling, the corresponding MEPs can be (have been) solved ab initio. Furthermore, the matrix elements are also computed in terms of the actual molecular states and the relevant coupling operators and not in terms of model potentials and adjustable coupling strengths. For example, see [ [18], which reports the calculation of total and of partial widths, obtained variationally (i.e., to all orders) of the predissociating math formulaexcimer state of HeF, which decays via radial and rotational couplings (computed from first principles) to the repulsive math formulaand math formulastates.

The common feature of electronic problems for both atoms and molecules is the orbital shell-structure represented by symmetry-adapted configurations and the requirement of adding correction terms representing EC. Conversely, for atoms, the number of practical possibilities of creating a variety of electronic structures that correspond to excited physical states is greater, as no open molecular dissociation channels exist. In many such cases, the role of EC is crucial for the calculation of small energy differences that determine whether a state is bound or belongs to the continuous spectrum as a resonance.

For example, the following situations involving exotic electronic structures, predicted from first principles using SPSQC, may look awkward, but are true: For ANIs, as excitation increases to even hundreds of eV, it is possible to construct wavefunctions for configurations with many open subshells. In such configurations, there is a unique decrease of screening, while the coupling of the many spins results in states whose maximum total spin is exceptionally high. The different self-consistent fields and the subtle differences of EC between the negative-ion state and the corresponding neutral state, may create conditions for the existence of highly excited ANI states that are bound, that is, they belong to the discrete spectrum of the negative ion, A, even though they are far inside the continuous spectrum, A + e.[19]

For example, helium in its ground state, 1s2 1S, does not bind an extra electron in a stable state, as is well known. Yet, when excitation of the three-electron system to 59.33 eV occurs, the bound state He 2p3 4So is formed. Similarly, suppose that an electron is “added” to oxygen atom and open-subshell configurations with maximum spin are considered. In this case, state-specific, many-electron calculations have shown that for an excitation of 600.6 eV above the O 1s22s22p4 3P ground state, the O 1s2s2p33s3p3 10S state is a bound, discrete state of the nonrelativistic Hamiltonian, lying inside series of resonance (Auger) states.[19]

Possible difficulties or shortcomings of methods used in CQC

The extensive application of methods of CQC, for example [ [4-8], has produced, and continues to produce, an enormous amount of information for molecular science. This production is facilitated immensely by the availability of flexible computer packages, written over the years by groups of collaborating experts, and by the existence of a countless number of molecular systems. As regards the frontier of contemporary CQC, the challenge seems to be along the direction of being able to handle progressively larger molecules, via the improvement of the methods and of the computational algorithms in conjunction with the continuing increase of computer power. Material science and even biology have a lot to benefit from reliable results of CQC in this direction.

Conversely, it is also useful from the point of view of basic science to acquire a better understanding of the fundamentals of electronic structures in ground and excited states, and of the degree of consistency of these popular methods in producing accurate results, and for which properties. Their comparative advantages and limitations are, of course, best understood by the practitioners. Nevertheless, based on the literature, it is reasonable to state here that, in general, their accuracy and degree of reliability depend on the system, even when the size of the calculation within the model is large. One set of systems where these methods (e.g., full configuration interaction (CI), coupled cluster, and DFT) may encounter difficulties and exhibit shortcomings is that of “states near threshold,” even when their energy is the lowest of their symmetry, as in cases of negative ions.

In support of the above statement, I direct attention to the following: The common characteristic of the conventional wavefunction methods of CQC is the use of large Gaussian basis sets, out of which the occupied and virtual atomic and molecular orbitals are constructed. The use of a single set of basis functions renders the algebra of these formalisms less complicated and their computational implementation more convenient. However, even for ground states of a given symmetry, such approaches may lose accuracy if the states are near threshold.

The root of the problem is in the function spaces used to describe the wavefunctions of interest, as in general, it is difficult to represent the mathematical properties of the continuous spectrum in terms of wavefunctions constructed out of the usual CQC basis sets. Furthermore, when the effective potential responsible for the bulk of the phenomenon is not Coulomb-like, as is the case of negative ions, the asymptotic part of the outer orbital acquires importance, and this means that it must be computed accurately, even if its contribution to the total energy is very small.

The following comment is relevant: the mere size of the conventional CQC wavefunction calculations usually leaves little doubt that the final result of a calculation is sufficiently accurate. This type of optimism can be found in many writings. Furthermore, it is often thought that a calculation on an atomic property by methods of CQC ought to be accurate, as for small values of Z, the number of electrons is most often smaller than that of the molecules where these methods are normally applied and produce energies of “chemical accuracy” (∼1 kcal/mol). However, one must not forget that the requirements on theory and computation for the reliable determination of properties of many-electron atoms, especially when excited states are involved, with zero order and correlation symmetry-adapted configurations having open subshells with orbitals of angular momentum reaching math formula, are different from those concerning the CQC-type calculations on most molecular ground states. Thus, even for small atoms, results of advanced methods used in CQC may sometimes differ substantially from each other.

As an example, I refer to Table 2.7 of [9], which compares results of four different calculations of the excitation energies of the low-lying valence states of beryllium with the experimental values. Three of them are in the CQC category. One set was obtained from the “full CI” calculations of Graham et al.,[20] who used a contracted Gaussian-type orbitals (GTO) basis (61 basis functions), and another set was obtained from the “full CI” and from a method called “spin-complete, spin-flip, CI singles” of Sears et al.,[21] who used a 6-31G basis set. Of these, only the full CI calculations of [20] can claim good accuracy when compared with experiment.

Conversely, the results of the fourth set of calculations were obtained with a minimal expense, following an analysis of the effects of EC on properties in the framework of the SPSQC.[9] Specifically, the calculations used only those wavefunctions and energies that are dictated by the “Fermi-sea” of each state. The orbitals are numerical, a prerogative that is applicable to atoms and diatomics. For the cases in question, the “Fermi-sea” superpositions consisted of only up to three symmetry-adapted configurations, with obvious advantages as regards transparency and economy. When the results from these simple calculations are compared to the experimental values, the comparison is quite satisfactory, and certainly more favorable than those from the CQC methods used in [21] (for a discussion of the state-specific “Fermi-sea” and its connection to the separation of EC into “dynamical” and “nondynamical,” see [ [9]).

The statements of the above paragraphs are linked to published results on atomic systems, where the electronic spectra are normally known with greater accuracy than for those of large molecules. In addition and in connection with the title of this subsection, I bring to attention the discussion and results of [ [22-24], where it was pointed out that certain CQC-type results led to wrong predictions, even qualitatively, for small energy differences near threshold (electron affinities) or for distribution of photo-absorption oscillator strengths over a section of the energy spectrum near threshold, in spite of the use of large basis sets and of extensive overall calculations with apparent good convergence with regards to total energies. The same conclusion applies to results of DFT calculations.

Prototypical samples of such cases and related discussions can be found in [22, 23(a,b)], where the conventional CQC methods whose results examined are the widely practiced full CI, B3LYP (DFT), and QCISD(T). Earlier,[24] the prediction which had been made, based on results of DFT calculations, that the noble gases, from He to Rn, have stable negative ions in their ground state, had been shown, theoretically and computationally, to be unfounded. Publication [24] was an early warning as to the possible difficulties that methods of DFT may face when tackling demanding cases of small energy differences, such as those of possible bound states of negative ions very close to threshold.

In conclusion, as discussed in [22,23(a,b)], for problems of many-electron states near threshold, in order for theory to be rigorous and computationally reliable, it must have a structure that optimizes the function spaces representing bound and free components separately, and a corresponding methodology for the calculation of the contribution of the continuous spectrum, for example, [9–11, 22, 23(a,b)].

Wavefunction Form and Function Spaces for Unstable States: Energy-Dependent Framework

The examples and discussion of the previous section underline the critical role that the form of the trial wavefunctions and the corresponding function spaces sometimes play when solving certain MEPs.

Particularly for the calculation of the various types of USCSs, the methods described in [ [9-11] are nonperturbative, and have a common feature of fundamental formal and computational importance: either in the time-independent treatments, or in the time-dependent ones, the sought-after solutions have the form of state-specific superpositions of bound and scattering, or scattering-like, components. The word “scattering-like” refers to complex functions representing the outgoing open channels of resonance wavefunctions, when these are calculated in the complex energy plane.

In all cases, these forms involve the continuous spectrum (multichannel in general), of the free-atom (molecule) Hamiltonian, whole explicit role in the pertinent physics cannot be ignored. Nor can the role of the electronic structures of various types and of the corresponding effects of ECs be substituted by simple models, as is sometimes done in various studies as a means of “demonstrating” the applicability of a particular formalism.

As regards the theoretical constructions for problems with USCS, it is possible to formulate them for atoms and for molecules based on the same fundamental principles and concepts. However, differences of physical and practical significance exist when considering the coupling operators, the different types of open channels and the appropriate functions spaces.

In the following paragraphs, I discuss aspects of the theory of field-free USCSs, which, on the one hand relate to fundamental properties and on the other justify N-electron methodologies for their calculation. In the Theoretical Time-Resolved Many-Electron Physics and Time-Resolved Signatures of Strong Correlations on Ultrashort Time-Scales sections, I will discuss briefly the time-dependent case of the USCSs created by the interaction with radiation pulses.

The foundations of the SPSQC to properties of USCSs were first published in 1972,[25] as part of a proposal for the unified many-electron treatment of arbitrary electronic structures, of low-energy scattering resonances (in negative ions they are also called compound states) as well as of high-energy excited states in the continuous spectrum (normally called autoionizing or Auger states). These were treated formally as “decaying” states, with complex energies defined as the complex poles of the diagonal matrix element of the resolvent operator, which drives the time-evolution from the initial state, Ψ0, which is a square-integrable many-electron wavepacket with real energy E0, to the continuous spectrum of the scattering states.

In the detailed discussion of [ [25] on the nature of the USCSs and on methods that had been used prior to 1972 for their calculation in atomic and molecular physics, the pertinence of the asymptotic boundary conditions was underlined. It was pointed out that, whether they appear as scattering resonances or as Auger states, “… from the mathematical point of view they do not have the useful property of square-integrability, having outgoing radiation boundary conditions (first introduced by Sommerfeld in a different context). Thus, they do not form a complete orthonormal set, and variational or perturbation theories dealing directly with such states must essentially be non-Hermitian in character” ([ [25], p. 2079).

Similarly, the description of formal properties was expressed in one of the statements as follows: “The wave function is now energy-dependent in the complex energy plane (to satisfy the physics of the uncertainty principle), and has a small continuum component, a fact which, in the time-independent equation (1), is expressed by the outgoing-wave boundary conditions of Eq. (2). This nonstationary process may also be thought of, in a time-independent sense, as a superposition of stationary ‘processes’ with different energies E. My choice to look at the physics as a time-irreversible process is rather arbitrary.” ([25], p. 2084). Here, the word “arbitrary” meant that the formal results for the computable energy and width are the same. Conversely, the time-dependent framework can penetrate deeper into the physics of such states.[10, 11]

The Eqs. (1) and (2) of [ [25] to which the above quotation refers define a time-independent many-electron complex eigenvalue Schrödinger equation (CESE) for the exact Hamiltonian, H, with outgoing-wave boundary conditions, based on the well-known 1939 result of Siegert[26] obtained from a model of s-wave scattering. Siegert's work showed, in accordance with the earlier Gamow assertion used in 1928 for the description of alpha particle decay that, assuming the formation at the energy position Er, with a narrow energy width, Γ, of an unstable state with a certain lifetime (inverse of Γ for exponential decay) the scattering solution has only outgoing-wave boundary conditions in the open channel(s), where the momentum is complex. By direct inference, this corresponds to a complex eigenvalue for the TISE (where now, due to the boundary conditions, the solution is outside the Hilbert space) given by zr = Er −  math formula. This is identical to the complex pole of the S-matrix, or to the aforementioned complex pole of the resolvent, which corresponds to math formula, with math formula =  math formula +  math formula, where math formula is the small energy shift due to the interaction of math formula with all the scattering states of the open channels.

The next two equations of [25], Eqs. (3) and (4), define and justify math formula, which represent the localized component of the exact eigenfunction on resonance, in terms of an effective multiparticle Schrödinger equation, math formula =  math formula, where math formulais the projection of math formula on math formula.

In order for either the S-matrix formalism, or the decaying-state formalism of [ [25], to be physically meaningful and computationally reliable, it was emphasized[10, 25] that it is crucial for theory to be able to demonstrate computationally the existence of math formulaas a valid solution of math formula =  math formula under conditions of square-integrability. This is where the interactions described by the operators in the Hamiltonian have their overwhelmingly large contribution. The effects of the residual ones contribute to the asymptotic part, Xas, of the exact wavefunction, which has been projected out in the definition and calculation of math formula. Xas acts as a small appendage to math formula, albeit with fundamental significance as its presence renders the exact state a member of the continuous spectrum, with a phase shift that has direct physical consequences.

The above brief description of the arguments in [ [25] and in many of our subsequent publications, is subsumed into the following statement that I quote from p. 187 of [ [10]:

Whether in wavefunction or in operator-matrix representation, the physically and computationally appropriate symbolic form that must transcend theoretical approaches to the understanding of resonance states is,

display math(1)

Depending on the formalism, the coefficient a and the “asymptotic” part, Xas, are functions of either the energy (real or complex) or of time.

As explained in detail in [ [9, 10] and their references, the function spaces representing the wavefunctions math formula and Xas are optimized for each system of interest, and the overall calculation of properties is carried out in terms of many-electron methods in frameworks that are Hermitian (time-independent as well as time-dependent treatments) or non-Hermitian (complex eigenvalues). In either case, the form (1) and the methodologies for its implementation allow keeping track of the effects of the various terms in a systematic way.

In conclusion, according to the proposals and formalisms discussed above, the understanding and calculation of properties of various categories of USCSs is achieved in a practical way using state-specific forms of wavefunctions, which facilitate the solution of the corresponding MEPs, while taking into account the effects of the continuous spectrum.

The generic-symbolic form (1) holds for field-free resonance states as well as for states dressed by a dc- or a periodic ac-field (treated as field-induced resonance states) with substantial differences, of course, in the nature of the coupling operators, in the number and types of open channels, and in the corresponding matrix elements.[10]

Resonance-state theory in terms of real and complex energies, and boundary conditions on function spaces

The polyelectronic theory of [ [25] emphasized the significance of computing accurately the square-integrable, math formula, and its real energy, math formula, before the necessary further steps that account for the contribution of the open channels to the properties of the resonance state. To justify its foundations, I considered the correspondence of resonances with complex eigenvalues in two ways.

The first way was by adopting the Siegert[26] outgoing-wave boundary condition at infinity in connection with the complex eigenvalue, and by asserting its applicability to a many-electron CESE, describing the state decaying by emission of one electron. (Eqs. (1) and (2) of [ [25]). Although neither the Siegert paper nor the subsequent general formal S-matrix resonance theory have anything to do with how to solve real MEPs while searching for the appropriate complex energies, this assertion set the stage for emphasizing the calculation of resonance wavefunctions in the form of Eq. (1).

The second way was the one which emerges from the implementation of the theory of decaying states, which is valid provided the many-electron wavepacket at t = 0, math formula, is computable from an effective equation, math formula =  math formula, with math formula math formula math formula. Accordingly, the solution which drives the time-evolution and defines the resonance state is a complex pole of the diagonal matrix element of the resolvent, math formula math formula math formula, ( math formula is a complex variable), that is, of math formula, in the lower half of the second Riemann sheet.

Apart from the mathematical consolidation of the concept of the USCS, the decaying state theory leads naturally to the support of the form of the solution as in Eq. (1), as the eigenvalue of the CESE corresponds to the complex energy solution of the defining equation,

display math(2)

where math formula is the complex “self-energy” operator of the autoionizing state and the functions on which it operates are square integrable. (See sects. 3.2 and 3.3 of [ [10]).

I now come to an alternative formal theory of resonance states, which is formulated exclusively in terms of real energies and is Hermitian. This was done in the elegant and pivotal 1961 paper of Fano[27] on autoionizing states and on the phenomenology of the photoabsorption cross-section in energy regions where these states exist. Fano did not deal with the “QC problem” of having to compute N-electron wavefunctions and properties. Instead, his formalism is based on the assumption that the relevant spaces are prediagonalized in some basis, as though the MEP has already been solved. Hence, the practical and significant question of how to obtain quantitative results for such states remained open, even though one can benefit greatly from the phenomenology, which is explicit in Fano's paper in terms of matrix elements.

Fano utilized and expanded expertly mathematics of the continuous spectrum, which had previously been introduced by Dirac, and solved formally for the energy-dependent mixing coefficients of the “discrete” and the “scattering” components of an autoionizing state. His theory never uses the notion or any result involving complex energies and complex eigenvalues for resonance states. The scattering formalism and its results are expressed on the physical axis of real energies.

So how can Fano's theory, which deals with the real-energy scattering Schrödinger equation, math formula =  math formula, with Dirac energy normalization, be reconciled with the definition of the resonance state in terms of the complex pole, which emerges in the energy-dependent scattering theory or in the time-dependent decaying state theory? Where is the complex eigenvalue hidden in Fano's theory and why? Can one take advantage of the explicit form of Fano's superposition in order to facilitate calculations of the energy and width of resonances in the complex energy plane?

When formulating the theory of [ [25], which, as I said earlier, considered as fundamental the fact that the resonance-state complex eigenvalue is associated with the dependence of the solution of the Schrödinger equation on the boundary conditions in the region beyond the extent of the square-integrable math formula, I was intrigued by the above questions, but I was unable to provide a rigorous answer. This was achieved in collaboration with Komninos and coworkers in 1980,[28] as part of our exploration of the use of complex functions, math formula, with complex coordinates, representing the asymptotic part Xas of Eq. (1) as math formula. The same approach has explained the presence of complex eigenvalues in field-induced resonances and the coordinate transformations which are required for their regularization.[29]

The essence of the argument[28, 29] is as following: Fano's real-energy superposition uses basis sets that consist of bound functions and of “standing waves,” which are real functions. In this way, the theoretical formulation is Hermitian, with the position, Er, and the width, math formula, given in terms of matrix elements that are defined for real energies. Conversely, by its very nature and definition, the decaying (resonance) state is represented by a wavefunction whose boundary condition at infinity contains only the outgoing-wave part of the standing wave. When the Fano solution is cast into a form depicting the separation of the standing wave into its ingoing-wave and outgoing-wave parts, and the coefficient of the ingoing wave is set to zero, explicit expressions for a complex asymptotic part of the resonance, Xas, and a complex energy, are obtained.

The analysis outlined above leads to a CESE where the Hamiltonian coordinates are always kept real and where the form of the resonance function is that of Eq. (1). In order for many-electron calculations to be possible in a practical sense, it is necessary to regularize the asymptotic part, Xas. This is easily done by making the coordinate of the outgoing-wave orbital, of each open channel, complex. The use of such a transformation for the regularization of the resonance eigenfunction, in the context of nuclear theory, where the potential is of short range, was first proposed in 1961 by Dykhne and Chaplik.[30]

When valid regularization procedures are applied to the asymptotic part of the wavefunction, Eq. (1), it is possible to develop practical many-electron methods for the calculation, to all orders, of energies and widths (partial and total) of atomic and molecular unstable states in terms of function spaces that are square-integrable. In particular, if this transformation is the easily implemented rotation of the coordinates of Xas into the complex coordinate plane, then the calculation can use the real many-electron math formula and a complex set of functions representing Xas, and search variationally for partial (and total) complex energies, optimized separately for each open channel.

Discussions on the theory and results supporting the statements of this subsection, together with many references, are presented in [ [10, 23, 25, 28, 29]

Finally, I point out that the concepts and the theory of field-free resonance states, which were outlined above, have been extended to time-independent nonperturbative treatments of field-induced resonances in the continuous spectrum, that is, to MEPs where the Hamiltonian includes either a strong static external field or a strong periodic ac-field, or both, in which cases the interaction of a state of the unperturbed Hamiltonian with the continuous spectrum results either in tunneling or in multiphoton (in general) transitions, or in both. In either case, the rates of transition are given by the imaginary parts (widths) of the complex eigenvalues of the corresponding many-electron, many-photon CESE, for example, [ [10, 17, 29]. Furthermore, energy shifts are obtained. In the framework of perturbation theory, the energy shifts are directly linked to the quantities known as static or frequency-dependent (hyper)polarizabilities. However, if the field is strong, the framework of perturbation theory breaks down, unless the method of computation goes to very high order. Such very high-order perturbative computations have not been carried out yet for states of atoms or molecules with more than one electron (see references in [ [10]).

On the MEP in the Theory and Calculation of Field-Free Resonances: A Brief Retrospection

The research activity in the theory and calculation of field-free resonances has been going on since the late 1920s to early 1930s, in AMOC physics as well as in nuclear physics, certainly not always at the desirable level of rigor, of significance of results and of computational utility. This is especially so when it comes to the possibility of handling correctly and efficaciously the many-body problem for arbitrary states and systems.

The discussion of this section presents some of its arguments in terms of brief citation and account of work on resonances of the smallest negative ion, H, where certain issues regarding the calculation of resonance states can be identified. Although the focus of the overall discussion concerning the state-specific theory is on the MEP for arbitrary electronic structures, the case of H allows some comparisons, as given that it is a system with only two electrons, it has been treated over many decades by different methods.

Usually, chronologically exposed accounts enhance the understanding of the significance of the introduction of new approaches and methods. Furthermore, there is often a general benefit from the careful study of systems with two or three strongly interacting electrons, as if the theory for their treatment is structured so as to be able to handle the MEP in terms of well-chosen zero-order and correlation function spaces, including the continuous spectrum, one may use the insight gained from these calculations for the execution of similar ones in larger systems, as the theory allows for the systematic calculation of the zero-order reference wavefunction regardless of electronic structure and of number of electrons.[9, 10, 25] For example, see the recent calculations and results on novel doubly excited states (DESs) of the 10-electron neon inside two-electron continua,[31] which were carried out in connection to the problem of understanding the “time delay” in the photoemission of the 2s and 2p electrons, as measured in the novel attosecond experiments of the Munich team led by Krausz and coworkers[32] and computed theoretically by the Athens team, in collaboration with Yakovlev and coworkers.[32]

To have a better appreciation of the significance of the above statements in the theory of unstable states, I start by recalling the following facts:

Because of EC, H has two bound states, (1s2 1S and 2p2 3P), and many resonances. Unique to H is the special class of “dipole” resonances, that were first studied and predicted formally in the early 1960s by Gailitis and Damburg, but which were comprehensibly computed and had their complete spectra analyzed quantitatively, including the effects of degeneracies and perturbations, only around 2000 by Bylicki and Nicolaides, via the implementation of the CESE theory, see [ [10] and references there in.

Hylleraas is best known in QC for his early accurate treatment of the two-electron problem in the ground state. I comment on two papers of his on H, one published in 1930 on the ground state, 1S, and the other published in 1950, on the “2s2p” 3P0 DES.

As explained below, it is a sign of the difference between the two types of states the fact that Hylleraas' treatment on the “2s2p” 3P0 DES was theoretically incomplete, with physically erroneous conclusions, as the continuous spectrum was not considered explicitly. The calculation was probably the first “large” CI treatment of an atomic energy spectrum, and used only bound functions (hydrogenic). The construction of the two-electron configurations was done with care to secure orthonormality to lower states of the same symmetry (see below).

I note that the literature on the theory and computations of resonance states for the period from the late 1920s to the early 1970s has been reviewed and/or cited in [ [10, 25].

Already in 1928, Slater[33] produced the first result of QC where a form of a correlated wavefunction is depicted analytically. Specifically, he concluded that, in the limit where the interelectron distance, r12, tends to zero, the solution of the TISE for the He 1s2 1S state must behave like math formula =  math formula, where math formulais the orbital product, math formula.

In 1929, Hylleraas published his extensively cited in the literature of QC variational calculations on the He 1s2 1S state, where the trial wavefunction contains terms that depend on the three coordinates, math formula, math formula, math formula. He found that convergence was accelerated dramatically.[34]

The analysis of Slater of the limiting forms of the eigenfunction for the two-electron 1S state, and the subsequent pivotal computational results of Hylleraas, are at the heart of the lore in much of the literature of QC on the MEP, namely, that the cusp behavior of the exact wavefunction as math formula is the reason for the slow convergence of CI calculations with orbital products (Slater determinants). In recent years, this understanding has led to the introduction into CQC of methods using “explicitly correlated wavefunctions.”[7]

CI wavefunctions with math formula-dependent basis sets were used in the 1960s for the calculation of doubly excited resonances in two-electron atoms, following the adaptation by O'Malley and Geltman[35] of Feshbach's 1958–1962 scattering theory for nuclear physics to two-electron systems. Given the size of the expansions that have been (are) used, such wavefunctions are indeed accurate and account reliably for EC. However, during the 1980s, results from state-specific calculations demonstrated that when it comes to such DES of He, whose wavefunctions (localized part), keep the electrons, on the average, at distances that are larger than those of the He 1S ground state, accurate results can be obtained with small expansions, where the zero-order is obtained as a state-specific, numerical multiconfigurational HF (MCHF) wavefunction and where additional configurations representing residual ECs are constructed in terms of variationally optimized virtual orbitals.

For example, see Table 2.4 of [ [9], where the SPSQC total energies of the first five 3,1D DES of He are compared with those obtained by Bhatia and Temkin from a CI calculation with 112 Hylleraas terms. These states are highly correlated, but the effects are taken into account efficiently by the state-specific approach without math formula-dependent terms, with expansions that were in fact smaller.

These results demonstrated that, although the standard argument found in the CQC literature, regarding the significance of the cusp, and, consequently, regarding the necessity of using expansions consisting of terms containing math formula coordinates for better convergence, is rigorous and computationally useful for compact wavefunctions of ground states, when it comes to the calculation of multiply excited states the degree of this necessity is diminished significantly, even when their localized wavefunctions, math formula, are highly correlated. Instead, the state-specific MCHF wavefunctions, if computed with numerical accuracy, provide a very good solution, already in zero-order, that is, before additional terms for “localized correlation” are included in the overall calculation. Of course, these terms may include math formula–dependent configurations if this is deemed absolutely necessary.

Conversely, in other approaches to the calculation of resonance states, the theory does not divide the function spaces into separately optimized parts, as does Eq. (1). Instead, it is based on the use of a single, square-integrable basis set, which must represent simultaneously the effects of both the bound and the scattering components. Such is the case of the calculation of the complex eigenvalues via the diagonalization of the Hamiltonian with complex coordinates, math formula, according to the “complex coordinate rotation” (CCR) method, which following the mathematical papers of Aguilar and Combes[36(a)] and Balslev and Combes[36(b)] on spectral properties of the Coulomb Hamiltonian, was initiated in the 1970s, for example [ [37, 38], and has since been studied and used extensively, for example, [ [39-41]. In this case, if results of high accuracy are desired, it is mathematically required that the set is near completion, and hence the consideration of math formula-type coordinates may be considered unavoidable.

For two-electron resonance states, the above requirement is met satisfactorily in many applications of the CCR method to isolated resonances with not too small widths. However, in the stringent test of the very weakly bound series of “dipole” resonances of H, even large CCR calculations with math formula-dependent basis sets have not been able to unveil high-lying members of the series. On the contrary, this was done successfully and uniquely via the CESE approach,[10] where there is separation and separate optimization of function spaces, according to Eq. (1) and in the spirit of the SPSQC. See Tables 4.2 and 4.3 of [10]. Furthermore, I note that the CCR yields only total and not partial widths. For the theory and calculation, to all orders, of partial widths of unstable states in many-electron systems, see [ [10].

I return to the work of Hylleraas on states of H. A few months after Hylleraas' publication on the variational calculation of the ground state of Helium, Bethe (1929) published a quantitative result that not only was computationally interesting but also had physical significance. It was probably the third such result of early QC. The previous two are those of Heisenberg (1926) on the “exchange force” and ortho- and para- Helium states, and of Heitler and London (1927) on H2. Specifically, Bethe predicted quantitatively the stability of the H 1s2 1S state via variational calculations with a three-term Hylleraas-type wavefunction. He found an energy that was 0.687 eV below that of the H 1s energy[34] (I recall that the energy of the “restricted HF” solution for the H 1s2 1S configuration is above the H 1s energy, and, therefore, it is EC that renders this state bound). In 1930, Hylleraas published the value 0.718 eV, obtained from a six-term expansion[34] (the accurate value, obtained theoretically as well as experimentally, is 0.755 eV).

Twenty years later, Hylleraas[42] turned to a computational exploration of whether the H 2s2p 3P0 state is below the H n = 2 level. Now, he used a different approach than that for the ground state, even though he treated the 3P0 state as though it is in the discrete spectrum. He somehow ignored the formal and physical role of the continuous spectrum defined by the open channel 1sεp 3P0, and after he obtained the results of his calculations (referred to below) he concluded that, “The existence of a second stable state of the negative hydrogen ion has been proved. It corresponds to the doubly excited (2s2p) 3P0 state of He with energy −1.506 in Rydberg units” (abstract of [42]). His conclusion regarding stability is erroneous, as the H 2s2p 3P0 state represents an unstable state, embedded into the 1sεp 3P0 continuum. In fact, there is indeed a second bound state of H below the H n = 2 threshold, but this is the 2p2 3P. However, Hylleraas concluded that, “…it is, however, doubtful whether the 2p2 3P state is stable at all for H….”

The 1950 paper of Hylleraas offers the opportunity for the following considerations. Conversely, the overall theoretical construction and discussion have the serious deficiency that they do not consider at all the possible effects of the continuous spectrum into which the H DES is embedded. The true nature of the state (resonance state) was conveniently ignored, and the calculation was set up as though it had to do with an excited discrete state. This calculation meant to incorporate effects beyond the simple approximation of using a single product of orbitals, which is reasonable only as a zero-order model. He did this by means of direct diagonalization of the two-electron H using a set of strictly bound two-electron configurations, math formula. In other words, the solution ψ(2s, 2p) was obtained as the superposition math formula. This is the well-known method of straight-forward CI, which has become one of the standard tools of “mainstream QC.”

Now, however, the state with which Hylleraas dealt has an infinity of lower states of the same symmetry. A brute force variational calculation with, say, a Hylleraas-type wavefunction, might in principle end up minimizing the energy up to the point where it represents, erroneously, a hydrogen atom plus a free electron of zero energy from threshold. Hylleraas addressed the question of orthogonality to lower states and argued that, by expanding the wavefunction in “some complete set of functions” and by “taking the second root in the resulting secular equation for the energy,” he could obtain a valid solution regardless of “whether the first root is a good approximation or not for the lowest state.” His choice of the first and second states was justified as follows: “The state is the second state in the series math formula 3P0, beginning with the ordinary singly excited state 1s2p 3P0” (I point out that there is no physical state of H to which a configuration 1s2p 3P0 can be assigned. This is simply the configuration that can be constructed using the discrete basis orbital functions with hydrogenic labels).

The diagonalization approach of using a single set of discrete functions in conjunction with the expansion form math formula, and statements of justificationthat are described above, constitute the immediate precursor of the “stabilization” methods for the calculation of energies of resonance states in two- and three-electron systems, which were applied and analyzed in the late 1950s and during the 1960s and early 1970s—see references in [ [10].

The recipe of the “stabilization method” for the calculation of energies of certain low-lying resonances, which was explained and advocated in that period, is simple and uses standard computational tools of QC. It requires the repeated diagonalization of H on a discrete basis, say math formula, in search of roots that stabilize as a function of some parameter. However, for arbitrary electronic structures and spectra, choosing suitable basis sets and searching in this way for all roots of the diagonalized Hamiltonian is not practical, while sometimes certain roots may be wrongly interpreted as representing real resonance states (see [ [10] for a discussion and references).

In contradistinction to the above, the approach that was proposed and demonstrated in [ [25] focuses on the direct solution of the state-specific wavefunction in zero-order and on the form of Eq. (1).[10] The problem of orthogonality to the infinity of lower states of the same symmetry was solved by reducing it to the requirement of imposing, when needed, orbital orthogonalities on a few HF and/or on virtual orbitals.[10, 25, 43]

The 1960s also saw the creation and quick development of formalisms, of penetrating analyses into the EC problem, of coded algorithms and of prototypical calculations which with the decisive help of evolving computational power and of large numbers of contributing scientists, eventually led to the explosion of CQC. These had to do with achievements in the following two domains:

  1. The theory that allows the consistent calculation of electronic structures at the HF (self-consistent field) level, and at the few-terms MCHF level, using the expansion method (analytic HF), was completed and programmed by Roothaan and his collaborators, and later by others. HF wavefunctions of ground state atoms and small molecules were published, as well as used, in analytic form, and computer codes for doing such calculations in a routine way became available. These facts constitute a turning point in the history of CQC, as they unleashed a huge potential for systematic computations of electronic structures.

    Parenthesis: concerning the theory developed in [25] and implemented to novel at the time-excited electronic structures, it is the adaptation of the code written by Roos et al.[44] that allowed the first demonstration of the possibility of solving variationally the HF equations for even weakly bound, multiply excited, open-shell structures such as 2s22p 2P0 and 2s2p2 2D in He.[10, 25] This demonstration was crucial in gaining trust in the validity of state-specific HF solutions (and later of MCHF ones, using Froese-Fischer's code[45] for the calculation of HF (or of MCHF) orbitals numerically) for highly excited configurations embedded in the continuous spectrum. The N-electron solutions are tested for the satisfaction of the virial theorem (constraint for localization) and for the correct behavior of the radials, including keeping track of their orthogonality to selected lower-lying orbitals.[10, 25, 43]

    Using state-specific MCHF solutions that are localized and numerically accurate as the zero-order reference wavefunctions, is an essential element of the SPSQC for the many-electron treatment of electronic USCS.[9, 10, 25]

  2. The first reliable quantitative results on EC for systems larger than He were produced, starting with the systematic and transparent 1960 CI work of Watson on Be. Formalisms that are relevant to electronic structures of atoms and molecules, and not just mathematical manipulations of basic elements of perturbation and variation theory for the real eigenvalue equation of Schrödinger, were introduced. These formalisms were the basis for the first steady steps in the understanding of EC in closed- or open-shell ground states of each symmetry, based on the definition, EC = exact nonrelativistic solution minus the result of the (restricted) HF calculation.

    I distinguish the contributions of, (I omit names of co-authors), Sinanoğlu, (closed- and nonclosed-shell many-electron theory), Kelly, (many-body perturbation theory), Kohn, (DFT), and Cizek, (coupled-cluster theory). By about 1969, their publications contained conceptual, formal, and/or numerical results, which made a positive difference as regards the pursuit of solutions to the problem of computing electronic structures and properties with EC. In turn, these had attracted the interest of other quantum chemists for further exploration, and for applications to larger and larger systems, including the possibility of drastic reduction in computational requirements of the standard method of CI, as was done in 1972 by Roos via his introduction of the “direct CI” method.

While the above serious progress in QC was taking place and the first results on properties of ground or a few low-lying excited states were being calculated with some of the effects of EC included, new experiments also initiated and/or perfected in the 1960s were creating new information on atomic and molecular spectra in different areas, which were then outside the formal and computational discussions that dominated QC. These had to do either with resonances in collisions of narrow energy beams of electrons with atoms and molecules, or with excitation into the continuous spectra by new sources of radiation, together with improved measurements of autoionizing (Auger) states. The results and the revealed possibilities of the new experiments could be interpreted as demonstrating that the unexplored realm of USCSs was open and rich with challenges to theory, especially as regards the proper treatment of the MEP in conjunction with the proper and tractable treatment of the effects of the continuous spectrum. In that respect, the publications that were already advancing “mainstream QC” were completely disjoint from the many-particle, many-channel quantum mechanics of the continuous spectrum, and in particular from formalism and from practical theory of electronic structures that are appropriate for the calculation of various types of low- or high-energy resonance (autoionizing) states in neutrals and their positive ions and in the distinct category of negative ions.

The state-specific theory of field-free resonances, initiated in [ [25] and further developed and expanded into other branches of the theory of USCSs,[9-11] was inspired by the above facts. Thus, its formalisms and methods aim at combining efficiently important and relevant elements of the mathematics and physics of the continuous spectrum with those that can handle the MEP in arbitrary electronic structures. To that effect, the basic tenet of the theory is the establishment of a reliable localized (square-integrable) component by first achieving a state-specific HF or MCHF solution (with the required orbitals characteristics and constraints—see [ [9, 10, 25, 45]) as a zero-order many-electron wavepacket inside the continuous spectrum. The remaining terms for the “localized” and for the “asymptotic” EC, are expressed by different function spaces and are computed by methods which are described in [ [10].

Theoretical Time-Resolved Many-Electron Physics

The interaction of atoms and molecules with radiation of all types produces a plethora of phenomena, which depend on the characteristics of both the material system and the radiation. Accordingly, there is a huge body of phenomenology based on models (e.g., two- or three-state systems) where the state functions or the time-dependent wavefunctions and the corresponding matrix elements enter only formally. For some types of models, numbers can be obtained as a function of parameters, which replace the necessity of ab initio calculation of the matrix elements in terms of which the full theory is expressed.

In fact, given a specific system, say a state of an N-electron atom interacting with a pulse with specified intensity, wavelength, and duration, the only way to produce reliable quantitative descriptions or predictions for the phenomenon or property under investigation is to create and/or apply formalisms and methodologies, which allow the ab initio computation and use of all the significant and physically relevant N-electron matrix elements.

In the pre-laser epoch, the only requirement in the theory of matter–radiation interaction for the calculation of N-electron matrix elements between correlated wavefunctions (energies and transition amplitudes) was that of calculating one-photon transition probabilities using the known expressions, valid for weak fields, for electric-dipole transitions (mainly) and electric-quadrupole transitions, for example, [46]. The first such N-electron calculations (for N larger than 2 or 3) with wavefunctions that include terms describing EC were published in the late 1960s and early 1970s.

In the post-laser epoch, many new and much more complex and challenging requirements were added, involving weak as well as strong periodic or pulsed radiation. At the same time, interest arose in the way atomic and molecular states respond to static, or nearly static, electric, or magnetic fields, which have large strengths.

The work that is briefly outlined in this and in the following section follows the research program, which we have been developing over the years regarding theory and methodologies for the ab initio solution of the two Schrödinger equations for stationary and nonstationary states and for the computation of various properties and phenomena, such as the ones that result from the interaction of radiation with many-electron systems.[9-11] To this purpose,

  • whether in one-photon or in multiphoton transitions or in field-induced tunneling,
  • whether the treatment is perturbative (case of weak fields) or nonperturbative (case of strong fields), and
  • whether the description of the phenomenon can be achieved in time-independent frameworks (static and periodic ac-fields) or requires a time-dependent one (short pulses of radiation),

the corresponding theories and computational schemes have been proposed and implemented with the aim of tackling efficiently the MEP.

Conceptually, formally, and computationally, the level of complexity and difficulty of the MEP for a Hamiltonian describing the system “atom plus external field,” differs from state to state in conjunction with the characteristics of the external field, according to combinations of the above categories.

As stated in section On the Domain and “Ages” of Quantum Chemistry, during the past three decades or so, significant advances in the production and spectroscopic use of strong and/or short to ultrashort (down to attoseconds), radiation pulses covering a very broad spectrum of frequencies, from IR to X-ray, have opened new scientific horizons.[12, 13] Among other things, they have created serious challenges to theory and computational methodology, in a new direction, which can be called “theoretical time-resolved many-electron physics” (TTRMEP).

The problems that are linked to TTRMEP have as their fundamental requirement the ab initio solution, math formula, of the appropriate for each case METDSE and the subsequent projection of math formula on physically relevant stationary states. By “many-electron,” I mean atomic and molecular states other than the singlet ground states of helium and of hydrogen molecule, for which the electronic structures are simple, and where direct one-electron photoexcitation, photoionization (without excitation–ionization of the second electron) involves only one channel with a one-electron core. I recall that, in almost all aspects of atomic and molecular time-independent or time-dependent physics, these two-electron states do not exhibit the richness of possibilities and of complexities that characterizes the spectra and properties of polyelectronic systems, even though certain interesting problems can certainly be constructed and solved to a very good approximation via their investigation.

As is generally accepted, given a particular problem and the characteristics of the radiation pulse(s) and of the state(s) with which they interact, the field intensities are heuristically divided into “strong” and “weak” (see classification above). In either case, to obtain quantitative information on various MEPs, it is necessary to construct and apply formalisms that can incorporate in a computationally tractable manner the details of electronic structures of the states that are significant for the solution of each problem.

Below, I outline the “state-specific expansion approach” (SSEA) to the ab initio solution of the METDSE, which is a nonperturbative theory, valid for both weak and strong fields. To place it inside a perspective regarding the early development and implementation of methods for the solution of such time-dependent problems in atoms and molecules, I start by referring to the “grid method.”

Many-electron systems in strong fields and the grid method

When the fields of the short pulses are strong, the METDSE must be solved nonperturbatively. This is equivalent to a calculation to all orders in perturbation theory, provided the series converges.

In the spirit of the time-independent approaches to the MEP that has to do with the solution of the many-electron TISE, the first important step toward the quantitative and reliable understanding of various properties and phenomena is the possibility of solving the METDSE approximately in the context of the independent particle model. Indeed, as the subject of strong-field physics was emerging experimentally, this computationally convenient approximation was first introduced and implemented in the late 1980s by Kulander and collaborators, as the “single active electron” (SAE) model, in a framework where the SAE TDSE is integrated on a “grid” of space-time points, for example, [ [47-49]. Although the method was formally restricted to be applicable only to the closed-shell single determinantal ground states of the noble gases, the pioneering calculations and analysis in that work provided significant initial information on magnitudes and on the phenomenology of the nonperturbative interactions of atoms with strong radiation pulses.[47-49] However, even in closed-shell initial states, such a drastic approximation, although it conveniently bypasses the requirements of the dynamic MEP, in general misses important information regarding the effects of ECs in bound electronic structures, of details of perturbed and unperturbed valence-Rydberg series, of multiply excited resonances, and of interchannel couplings in the continuous spectrum.

More than 25 years after the first implementation of the grid method for the treatment of problems of time-resolved many-electron physics, its formal and computational extension beyond the SAE remains a desideratum for many-electron systems, even when the initial state is represented by a closed-shell wavefunction. This fact has led some researchers to opine that the METDSE is untreatable “in the foreseeable future,” apparently linking its solution to the formal and practical applicability of a grid method. To be specific, I quote from the paper by Maquet and Taïeb[50]: “In principle, the experimental data could be retrieved by solving the TDSE for the atomic system, in the presence of the two pulses. However, the precise calculation of the response of a rare gas atom presents considerable difficulties. This is because the numerical resolution of the TDSE for a multielectron system remains well beyond the capability of present-day computers and that no satisfactory solutions will be available in the foreseeable future. In fact, present state-of-the-art computations rely on the single-active electron approximation, with model potentials that permit one to reproduce the bound state spectrum of the atom with a satisfactory accuracy” ([ [50], p. 1850).

Strong field and the SSEA

The above statement, made in 2007, would have been true before 1994, which is when an alternative approach to the nonperturbative solution of the METDSE, the SSEA, was proposed and demonstrated on the four-electron negative ion of Li.[51] The SSEA, which is outlined below, has since been applied successfully to a variety of prototypical dynamic MEPs, some of the most recent ones being,

  1. The first demonstration of coherent excitation, formation, and decay of channel-dependent inner-hole resonances in the 13-electron Aluminum.[52]
  2. A pump-probe with time delay scheme for the excitation of the rather exotic triply excited bound state, He 2p3 4So, via an intermediate resonance state.[53]
  3. The first theoretical demonstration of the existence of the atomic-dynamics-dependent time delay in the photoionization of the 2s and 2p electrons from the 10-electron Neon.[32]

The SSEA to the ab initio nonperturbative solution of the METDSE was first formulated and implemented in 1993–1994.[11, 51] It is a wavefunction method, which does not require the use of a space-grid or of any type of “absorbing potentials.” It is based on the standard principle of quantum mechanics, namely, that of the expansion of a wavepacket in terms of the stationary states of the unperturbed system, including, of course, those of the continuous spectrum. The mixing complex coefficients are time dependent. Accordingly, the form of the solution of the SSEA is (I omit the index for each possible channel)

display math(3)

The expansion (3) holds for atoms as well as for molecules. When implemented, its level of accuracy depends on the relevance of the states math formula and math formula to the problem under consideration and on the degree to which the trial wavefunctions represent the exact ones.

For many-electron atoms, the state-specific bound wavefunctions, math formula, and the energy-normalized scattering wavefunctions, math formula, are computed in a manageable way for each bound electronic structure (discrete states as well as the localized part of resonances) and for each value of the energy in the continuous spectrum, whose threshold is set at zero. The scattering functions can be computed numerically in a term-dependent core potential.

For diatomics and problems of multiphoton dissociation, there is no MEP after the construction of the one-dimensional potential curves. Hence, in this case, for a smooth continuum where no other electronic curve mixes with it, it is computationally expedient to use a single basis set for both the bound and the free vibrational states.[11]

For polyatomic molecules, the nonperturbative solution of the METDSE according to the expansion (3) is an open field, as the reliable and systematic calculation of excited states, of resonances, and of pure scattering states remains a desideratum. Of course, for problems that are simplified, expansions which approximate that of Eq. (3) can always be adopted. However, their degree of reliability will be limited, unless the contribution of the continuous spectrum is taken into account consistently, and in a way which is mathematically justifiable.

In addition to the issue of the degree of accuracy of the wavefunctions entering in expansion (3), there is the issue of the calculation of the time-dependent coefficients. For weak fields, it is reasonable to expect that a framework of lowest-order time-dependent perturbation theory suffices. However, for strong fields, the calculation must be nonperturbative, via the solution of the corresponding coupled equations, which is what the SSEA does. In the case of a time-dependent Hamiltonian, math formula =  math formula +  math formula, where an atomic (molecular) state is perturbed by one or more time-dependent radiation pulses with interaction symbolized by math formula, the necessary input for such a SSEA calculation are the state-specific bound–bound, bound–free, and free–free matrix elements of math formula in addition to energies.

By its very construction and computational methodology, the SSEA allows the direct monitoring of the relative significance of each state in Eq. (3). The type, quality, and number of the wavefunctions are determined by each problem of interest, depending on the interplay between the characteristics of the radiation pulse [incorporated into the perturbing operator, math formula] and the spectrum of the atom (molecule), or of their positive or negative ions.

The optimal calculation of the above coupling matrix elements is achieved using wavefunctions that are as state-specific as possible.[9-11] To this purpose, we have shown how to apply methodologies that allow the computation of state-specific electronic structures of any type of zero-order single, or multiconfigurational label. Depending on the electronic structure and on the type of problem, the calculations aim at the solution of HF or of MCHF equations, or of MCHF plus the remaining important ECs, computed variationally to all orders. The continuous spectrum is represented rigorously by energy-normalized scattering orbitals obtained numerically (often aided by the analytic asymptotic formulae of WKB theory) in fixed term-dependent potentials of polyelectronic states. These represent the smoothly varying continuous energy spectrum with zero intrachannel coupling.

Weak field and analytic results

The nonperturbative SSEA, which entails the numerical solution of thousands of coupled equations mainly because of the large number of energy-normalized scattering wavefunctions, is systematic, generally applicable, and transparent. However, it is also computationally intensive and its requirements are much more demanding than those of a lowest order perturbative treatment. In the former case, all types of states that may be coupled via the total Hamiltonian must be considered, although a careful examination and the tracking of convergence eventually separates out a subset that makes the overall calculation tractable. In cases of problems involving weak fields and short time durations, lowest order perturbative approaches are perfectly valid. Such approaches may lead to useful results, which are analytic and easily applicable, involving only a very small number of N-electron matrix elements between the states, which are directly involved in the scheme under consideration. For example, such results were derived for the description and easy computation of the time dependence of the formation of isolated resonance states by a short pulse[54, 55] or of a case of pump-probe excitation of multiply excited states.[53]

Time-Resolved Signatures of Strong Correlations on Ultrashort Time Scales

The standard treatment of problems of phenomenology and of computation in AMOC physics uses stationary state formalisms, and this is in agreement with corresponding experiments, even in cases where one thinks of certain phenomena as results of an evolution in time. For example, such formalisms define the energy widths of resonance (autoionizing) states or the rates of transitions of various processes as a function of energy. Indeed, when the measured quantities represent some type of an average over time, the energy-dependent formalisms yield the same or equivalent results as time-dependent formalisms, thanks to Fourier transformations.

Conversely, it is possible to define and explore, theoretically and experimentally, time-dependent processes that are described exclusively in terms of nonstationary states, where the system has not yet “equilibrated” or measured in stationary states. Their proper description requires, apart from phenomenology, the solution of time-dependent MEPs, that is, the solution to a good approximation of the METDSE, math formula, and its subsequent use, depending on the problem. In this regard, experimental science has made impressive progress toward the possibility of measuring events, that is, taking snapshots, of processes occurring on the scale of a few femtoseconds and, since 2001–2002, even of attoseconds.[12, 13, 56-58]

The present section brings together the theoretical and computational notion of “EC” and of “CI,” which are at the heart of QC, with the dynamics of high-energy excitation of resonance (autoionizing) states (including, of course, the effects of the scattering states of the background) in the context of new theoretical and experimental possibilities for obtaining quantum information on the time-axis at ultrashort (femtoseconds and attoseconds) time-scales. This type of time-resolved information on microscopic processes cannot be Fourier related to quantities defined on the energy axis. It represents unique and novel physics and has opened new horizons for time-resolved spectroscopy and control of electronic and nuclear “motions.”

The early theoretical and experimental results on the details of attosecond time-resolved processes involving electron rearrangements in unstable states

When thinking of, say, an autoionizing state, even though the implied decay is time-dependent, the intrinsic property of energy width, math formula, which is the inverse of the lifetime as defined by exponential decay, is deduced from a measurement on the energy axis. Conversely, a fuller picture, with more possible insight, is obtained when the METDSE is solved from first principles, provided a well-defined wavefunction for the initial state, math formula, is available. To obtain time-resolved information on the decay of real, multielectron unstable states, at very short and at very long times, that is, outside the realm of exponential decay, in 1996 we introduced and demonstrated the methodology of the SSEA to the calculation of the time-resolved decay of unstable states that are very close to threshold, of neutral atoms and of negative ions.[10, 59, 60] The cases studied involved processes of barrier penetration for a shape resonance (e.g., the three-electron He 1s2p2 4P) as well as of two-electron rearrangement, which normally dominates nonrelativistic or relativistic autoionization (e.g., the 20-electron Ca KL3p63d5p 3Fo).

On the experimental side, the possibility of time-resolving the autoionization process during core-hole relaxation in a real system constitutes the first application to a physical problem of the then emerging spectroscopy involving single attosecond pulses. Specifically, in 2002, Drescher et al.[58] used a novel source of EUV attosecond pulses and, using pump-probe techniques with ultrafast sampling of electron energies, recorded the exponential decay of 3d holes in Krypton and deduced a lifetime of 7.9 ± 1.0 fs. As the authors[58] pointed out, this value corresponds to an Auger width of 84 ± 10 meV, in agreement with a previous measurement in the energy domain, which gave 88 ± 4 meV.[61]

Attosecond time-resolved effects of CI in doubly excited and inner-hole states and the projection of the “motion” of electrons

The decay of an unstable state is understood mathematically and physically as starting from the square-integrable ( math formula), which, unlike the physics of the decay of excited states by photon emission (Wigner–Weisskopf theory), is not an eigenfunction of the exact Hermitian Hamiltonian, math formula, whose stationary states form an absolutely continuous energy spectrum above the corresponding fragmentation threshold.[10, 25] This fundamental assumption implies that the state has lost memory of its preparation step.

In 2002, a few months after the announcement of the controlled production of attosecond pulses,[56, 57] we introduced the proposal, with supporting quantitative evidence that the prospects were favorable for spectroscopy on the attosecond scale that allows the time-resolved measurement of phenomena, which are caused by strong ECs and corresponding CI in inner-hole and in multiply excited states, normally found in the continuous spectrum. This proposal and corresponding results solved the time-dependent coherent dynamics of excitation and decay (for reasons of efficiency we used two excitation pulses) of highly excited unstable states whose wavefunctions are highly correlated.[62, 63]

Specifically, the math formula was calculated via the SSEA as the nonperturbative solution of the METDSE describing the coherent excitation by ultrashort pulses and the concurrent decay of DESs of He, in transitions from the ground, 1s2, or the metastable, 1s2s, 1S states, to DES of 1P0 symmetry labeled by the “2s2p,” “2p3d,” and “3s3p” configurations. I emphasize that the scattering states were present explicitly in the expansion of Eq. (3).

The results and their analysis in [ [62, 63], provided the first insight on time-resolved effects of strong ECs that may take place on the attosecond time-scale. For example, we determined how the correlating superposition of configurations labeling DES in that energy region can lead to ultrafast intra-atomic rearrangemets of pairs of electrons representing different electronic “geometries.” Assuming that this time-dependent superposition can be probed with attosecond speed, by projecting the wavepacket represented by math formula onto stationary states of the continuous or the discrete spectrum, the probability of measuring final states will depend on the dominant configurations and, therefore, reveal the electronic geometry. It is possible that if such situations were further explored and understood in practice, they could prove useful to future techniques that could trace “motions” of electrons in molecules.[62, 63]

Our 2002, 2004 publications[62, 63] and the choice of excitation-decay schemes in Helium as testing ground of theoretical models for the study of issues regarding time-resolved processes in experiments using ultrafast pulses of high frequencies, were the first, following the publication of the experimental demonstration of attosecond pulses,[56, 57] to orient theoretical and experimental research toward this system and toward issues regarding the concept and demonstration of effects of time-resolved strong ECs and CI in connection with attosecond pump-probe spectroscopy. It has been followed by additional exploration of a similar nature on the same system by us, for example, [ [54, 55] as well as by other research groups, for example, [ [64-66]. The resulting information keeps enriching knowledge on new processes, setting the stage for the eventual possibility of analyzing, tracing and controlling the “motion” of electrons in large systems.

In our case, the work reported in [ [54] refers to the computation of the first “pictures” of the time-dependent formation of the profile of an autoionizing state upon excitation by an ultrashort pulse, and its possible dependence on the pulse duration. This work, which constitutes a clear example of time-resolved effects of ECs involving the continuous spectrum, is summarized below.

Time-dependent formation of the excitation profile of an autoionizing state on ultrashort time scales

In 2005, Wickenhauser et al.,[67] using a model system, which assumes data from (super)Coster–Kronig transitions with lifetimes of ∼400 as, explored properties linked to the aforementioned wavepacket, math formula, from a different angle. They “investigated the feasibility of observing the buildup of a Fano resonance in the time domain by attosecond streaking techniques,” (abstract of [ [67]), and produced results on the phenomenology of such a possibility.

Their publication led us to return to the problem of the ab initio solution of the METDSE for the Helium system to produce, for the first time, quantitative results for the time-dependent formation of the profile of the He 2s2p 1P0 resonance state upon excitation by a well-specified ultrashort pulse.[54, 55] This entailed the implementation of the SSEA to determine how the effects of the correlations of the pair of the initially bound electrons evolve and eventually form the stationary superposition of bound with scattering components on the energy axis that gives rise to the asymmetric Fano profile. In addition, using first-order time-dependent perturbation theory, we derived and applied an easily applicable analytic formula for this phenomenon, valid for weak pulses. It was shown that, for weak fields, its results indeed coincide with those obtained from the nonperturbative solution of the METDSE (Fig. 2 of [ [54]).

Figure 1 shows the time-resolved differential ionization probability, math formula, for the gradual formation of the asymmetric profile of the of the He 2s2p 1P0 resonance state upon excitation from the ground state by a sin2 pulse of duration 450 a.u (11 fs) and of energy of about 60.1 eV (the figure is taken from our work in [ [54]). math formula is defined by the projection of math formula on the scattering stationary state at energy math formula, math formula math formula math formula, where math formula is the energy above the ionization threshold. As the two electrons in the bound orbitals 2s and 2p are allowed to correlate, and localized as well as asymptotic components are added, the quantity math formula dominates the transition of the two-electron wavepacket to the continuum of the scattering states in the region around math formula, whereas interference effects eventually stabilize after about 180 fs, when the completion of the formation of the stationary superposition of the resonance takes place.

Figure 1.

Time-resolved differential ionization probability, math formula, for the gradual formation of the asymmetric profile of the He 2s2p 1P0 resonance state upon excitation from the ground state by a sin2 pulse of duration 450 a.u (11 fs). ε is the energy above the ionization threshold. After about 180 femtoseconds, the characteristics of the curve, (the position, the Fano q parameter and the width), are the same as the ones that have been obtained from various treatments on the energy axis. (Reproduced from Ref. [ [54], with permission from

©American Physical Society


Figure 2.

Sketch of the overall discussion and argumentation of the paper, as regards the issue of the “ages of QC.” The solution of time-independent or time-dependent MEPs involving field-free or field-induced unstable (nonstationary) states relaxing into the continuous spectrum, requires the combination of versatile N-electron methods of excited electronic structures with appropriate theory that accounts for the distinct characteristics of the continuous spectrum and of the interaction of matter with strong static, periodic or pulsed (mainly) fields, or with hyper-short radiation pulses.

As a proof-of-principle demonstration of the same theory and computational methodology to a many-electron, multichannel system, in 2007, we also published results similar to those of the previous applications to Helium, to the 13-electron Aluminum.[52]

I close by pointing out that the analytic formula derived (corrected for a misprint) and used in [ [54], can be used for appropriate predictions in many-electron atoms, provided its parameters are known from computation or experiment. For example, its application shows that Auger states of atoms in the middle of the Periodic Table, whose width is sufficiently large, about 2 eV or more, are suitable candidates for observing the formation of their stationary profile with attosecond duration.


The essence of the discussion, results, opinion, and supporting information contained in this article can be summarized as follows:

Over many decades, an understanding has been molded that QC is represented and characterized nearly exclusively by the activity and progress in the direction and methods of CQC. As it follows from a perusal of voluminous work on formalisms and on applications, the focus of CQC is on the accurate calculation of “energetics” and of other properties of ground states (including those of a few low-lying discrete excitations) in atomic and molecular (mainly) systems with few or with many electrons, via the solution of the time-independent, real-eigenvalue, Schrödinger equation, for example, [ [1-8]. The achievement of this goal entails the possibility of tackling efficiently the MEP, for nonrelativistic as well as for relativistic field-free Hamiltonians. In certain problems, the resulting accurate potential energy surfaces are used for calculations that provide information on simple chemical reactions or on ro-vibrational spectra of ground or low-lying excited electronic states. Based on the above understanding, four “ages of QC” have been defined and quoted in the literature.[2, 3]

Conversely, given the various experimental possibilities of excitation and measurement of energy-dependent effects associated with resonances, or of time-dependent processes involving atomic and molecular states in the continuous spectrum, certain wide-ranging themes ought to be considered as defining “ages” in the development of QC, in addition to those that have been attributed to the progress that has been achieved in CQC, which has become, de facto, “mainstream” QC. The Hamiltonians determining the physics in such themes describe either the free system or include the interaction with external static or frequency-dependent electromagnetic fields. In the latter case, the challenge to modern QC comes from the need to combine methods for the accurate solution of open-shell electronic structures and the MEP with advanced theory of the physics created by the interaction of ground or excited states of many-electron systems with strong fields (periodic or, especially, pulsed).

In section On the Domain and “Ages” of Quantum Chemistry, four topics categories were singled out, which, individually or in combination, ought to qualify as indicating additional “ages” of modern QC, some of them very young and others somewhat older.

In this context, I gleaned elements of theories and state-specific many-electron methods of computation of such states and related phenomena, which have been proposed and implemented by the author and his collaborators over a few decades.

It is emphasized that the article is not a general review. In particular, the selected examples and corresponding references that are discussed in the Many-Electron States near the Fragmentation Threshold Perturbed by the Continuous Spectrum and the on the MEP in the Theory and Calculation of Field-Free Resonances: A Brief Retrospection Sections, including that of the pioneering attempt of Hylleraas[42] to investigate the H 2s2p 3P0 doubly excited resonance state via the diagonalization of a suitably constructed energy matrix, are chosen with the purpose of supporting the arguments and explanations.

I focused mainly on the following two themes:

  • 1. The nature and the calculation of resonance states in time-dependent and time-independent frameworks.[10]

By going back to the theory of [ [25], it was outlined how the consideration of boundary conditions sets the stage for the explanation of the fundamental properties of such states. It was explained again, how the choice of the form of the resonance eigenfunction, Eq. (1), which focuses on the significance and reliable calculation of the square-integrable many-electron wavepacket, ( math formula), describing the state at t = 0, is crucial for the economic and reliable solution of the Schrödinger equation, either for real energies or for complex eigenvalues. It was outlined how the CESE, where the exact form of the complex eigenfunction is that of Eq. (1), has been derived from the real-energy scattering wavefunction of Fano's[27] Hermitian theory. The complex eigenvalue is obtained via the variational solution of the CESE, but it can also be computed via Eq. (2), which results formally either from energy-dependent scattering theory or decaying-state theory.[10, 25]

The separation (1) holds for real energies and for a many-electron treatment using “multichannel reaction matrices.” Alternatively, it justifies and allows the efficient calculation of energy partial and total widths and shifts to all orders, with interchannel coupling, in terms of nonHermitian constructions, which use Hamiltonians with real coordinates and square-integrable function spaces consisting of real basis sets for the bound orbitals and of complex sets (using complex scaling) for the open channel orbitals of the asymptotic correlation functions.[10]

  • 2. The possibility of identifying and computing effects of time-resolved strong ECs and CI in dynamic phenomena that can be generated in pump-probe experiments using radiation pulses with duration in the time-scales of a few femtoseconds or, more pertinently, of attoseconds. The solution of the METDSE, whose knowledge and use are required for the quantitative understanding of such problems, is achievable in terms of the SSEA.[9, 11] The structure of the SSEA is such that it allows the use of efficient methods of computation of bound, of resonance, and of purely scattering states that must be included in Eq. (3), for each problem of interest.

Reviews of the theories and computational approaches to the solution of MEPs such as the above, accompanied by sample applications, can be found in [ [9-11]. Here, I chose to present the quantitative results, first published in [ [54], on the time-dependent formation of the asymmetric profile of the well-known He 2s2p 1P0 resonance upon its excitation from the ground state by a femtosecond pulse (Fig. 1).

In conclusion, thanks to a variety of new experimental possibilities, a broad horizon for QC is open in the direction of treating dynamical problems of electronic excitation and mixing with the channels of Rydberg series and of scattering states, of N-electron excited states near and, mainly, above fragmentation thresholds. The Hamiltonian can be either field-free or may include the interaction with weak or strong static, periodic, or pulsed fields. In the case of strong fields, the overall treatment must account for effects of higher than the lowest order of perturbation theory. In practice, this requirement is satisfied by implementing nonperturbative approaches. In all possible types of corresponding time-independent or time-dependent problems, the theoretical constructions and the computational methodologies must be capable of tackling in a rigorous, yet tractable way, the effects of highly excited electronic structures, of ECs and of the continuous spectrum, in conjunction with the effects of the external fields.

A sketch of the overall discussion and argumentation of the paper, as regards the issue of the “ages of QC,” is depicted in Figure 2. The previously proposed four ages of QC[2, 3] are associated with progress in the achievements and capabilities of CQC toward the quantitative study of properties of ground states (mainly) and of low-lying discrete excitations along the direction labeled “size of N-electron system.”


  • Image of creator

    Cleanthes Nicolaides was born in Athens, Greece, on December 31, 1946. He obtained his Ph.D. in Theoretical Chemistry from Yale University, June 1971. His academic career started at Yale as a Lecturer and Assistant Professor. In 1976, he was appointed the founding Director of the Theoretical and Physical Chemistry Institute (TPCI) in Athens, a position he held until 1994. He is retired Professor of Physics, National Technical University, Athens, and Director of Research, TPCI, emeritus. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]