## 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, H_{2}O, or CH_{4}. 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 H_{2}, 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:

- 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.
- 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.
- 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.
- 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.