Found Chem DOI 10.1007/s10698-012-9170-0

Beyond the orthodox QTAIM: motivations, current status, prospects and challenges Shant Shahbazian

 Springer Science+Business Media Dordrecht 2012

Abstract Recently, the author of this paper and his research team have extended the orthodox quantum theory of atoms in molecules (QTAIM) to a novel paradigm called the two-component QTAIM (TC-QTAIM). This extended framework enables one to incorporate nuclear dynamics into the AIM analysis as well as performing AIM analysis of the exotic species; positronic and muonic species are a few examples. In present paper, this framework has been reviewed, providing some computational examples with particular emphasis on origins and applications, in a non-technical language. The main questions, enigmas and basic ideas that finally yielded the TC-QTAIM are considered in chronological order to help the reader comprehend the intuition behind the math. Finally, it is demonstrated that the TC-QTAIM and its more refined versions are able to tackle problems inaccessible to the orthodox QTAIM. Keywords Quantum theory of atoms in molecules  Born–Oppenhiemer approximation  Clamped nucleus model  Exotic species  Non-adiabatic wavefunctions ‘‘There must be chemistry in all these wave functions because we live in one world only’’ Paul Popelier (Popelier 2000).

The road map In formal scientific writings, the authors usually just state the final results disclosing the completed construct, leaving out the motivations and the history of evolution of the relevant ideas. The opportunity to state the real history is rare. Accordingly, I want to use this opportunity provided by the Foundations of Chemistry to disclose the history and ideas This paper is dedicated to the memory of Prof. Richard F. W. Bader (1931–2012), a true genius and one of giants of theoretical chemistry. S. Shahbazian (&) Department of Chemistry, Faculty of Sciences, Shahid Beheshti University, G.C., Evin, P.O. Box 19395-4716, 19839 Tehran, Iran e-mail: [email protected]

123

S. Shahbazian

behind the newly developed two-component quantum theory of atoms in molecules (TC-QTAIM); a novel paradigm that aim to extend the atoms in molecules (AIM) analysis to unorthodox domains. Inevitably, this is a personal presentation neglecting the usual standards of a formal scientific paper. I hope that this informal view act as a supplement to our recent technical contributions (Goli and Shahbazian 2011a, b, 2012; Nasertayoob et al. 2011; Heidar Zadeh and Shahbazian 2011). The QTAIM is the theory of form, structure and properties of AIM, the chemical atoms (Bader 1990; Popelier 2000; Matta and Boyd 2007). Although struggles and controversies over its nature and position in modern quantum chemistry are yet ongoing, there is no doubt that the QTAIM is in its growing age. A survey on current computational literature demonstrates that its use as a tool to extract chemical observables from intricate ab initio wavefunctions, rationalizing bonding and reactivity in computational realm, is ubiquitous. Beyond computational realm, crystallographers’ community has been also tremendously attracted toward the QTAIM in the previous decade witnessing a modern age of charge density analysis (Gatti and Macchi 2012). Are all these advances the whole story? In forthcoming sections it is demonstrated that this is just the start of a new age and we have discovered only the tip of an iceberg. The orthodox QTAIM with all its pros and cons is just the simplest version of a hierarchy of AIM methodologies. This extended hierarchy set the stage for unorthodox applications and reveals delicate issues yet considered by many chemists beyond the realm of chemical thinking. In the section entitled ‘‘Motivations: Why seeking further?’’ the main motivations that prompted our research group to develop an extended framework encompassing the orthodox QTAIM but going beyond its orthodox domain are described. The main enigmas and questions are discussed and it is shown that there were clear signs, recognized even by the late Richard Bader, that an extended QTAIM is needed to answer some basic questions and unresolved issues. Then, in the section entitled ‘‘Current status: What we know and what remains to be known’’ employing a simple language (taking seriously the objection of the intricacy of the used mathematical language in our previous papers stressed by some reviewers) the main aspects of the extended formalism are discussed. Accordingly, the goal of this section is not a rigorous presentation of the material found in the original literature, but an illustrative discussion by considering some simple examples. The ongoing projects are also disclosed briefly to make a complete picture of the fabric of the extended QTAIM; the recent novel concepts and predictions emerging from this framework are also briefly sketched. Subsequently, in the last section entitled ‘‘Prospects and Challenges’’ the future plans are disclosed. I hesitate entering the realm of wild speculations; the proposed developments are those that are now under serious consideration in our group and there are good reasons to believe that they will be materialized in near future. I hope this informal presentation assists grasping the main ideas and key features of our proposed extended QTAIM that are somehow buried, albeit inadvertently, in the technicalities of previous contributions.

Motivations: why seeking further? Why seeking further? Are there signs that the orthodox QTAIM is missing something? Well, it seems there are clear hints of some intrinsic shortcomings. In the forthcoming subsections, some fundamental issues are presented that the orthodox QTAIM is unable to deal with. Some have been noticed by others whereas some issues do not seem to be considered or even noticed seriously.

123

Beyond the orthodox QTAIM

Robustness of noble reality: a guiding principle Reality is always ‘‘created’’ within a ‘‘context’’; context independent reality is just a myth. In science, contexts are theoretical frameworks, an interconnected network of concepts, mathematical entities and relationships. An entity is called real since it is located in a fabric of well-structured network of concepts and relationships. This is why we believe that the mass, energy, electric charge, electric and magnetic fields or temperature are real entities; mechanics, thermodynamics and electromagnetics act just as proper contexts. We call a theoretical construct a local reality if it emerges from just a certain context. However, it sometimes seems that there are concepts emerging from various contexts; energy is just as an illustrative example. It is defined and introduced in mechanics, thermodynamics and electromagnetics as classical contexts nevertheless also appears in quantum mechanics as well as special and general relativity as modern contexts. Accordingly, it is one of those entities that have been transferred from classical to modern physics. We call such theoretical constructs emerging from various contexts noble realties since they survive upon changing or ‘‘widening’’ the contexts; like a transition from mechanics to thermodynamics or the revolutionary transition from classical to modern contexts. One may call this trait of noble realties robustness. A robust entity survives or better to say persists, albeit sometimes by some ‘‘deformations’’, in a wider context. Space and time are absolute and passive in classical mechanics, whereas in the general relativity, they persist as fundamental concepts, however they are now dynamic entities influenced by the matter. The same classification is also applicable in quantum chemistry. As an illustrative example, in non-relativistic quantum chemistry, contexts are Hamiltonians and corresponding wavefunctions. Mean field theories, Hartee-Fock (HF) or Density functional theory (DFT), create their own conceptual constructs, thus one-electron functions emerge that are local realities, however not all constructs survive in a transition to high level correlated methodologies; they are not noble realities (Shahbazian and Zahedi 2006, 2007). What about the concept of an atom in a molecule? Well, in the orthodox QTAIM as well as within the context of its rival theories AIM are local realities. However one may ask: Are AIM noble realities? Based on previous discussion, this fundamental question may be replaced with another equivalent question: Are AIM derivable from various contexts? The formalism of the orthodox QTAIM is based on two tenets: I. The one-electron density is the basic function that one must seek for the structure and form of molecules. In other words, electron density dictates the morphology of AIM, topological atoms and molecular graphs, as the underpinnings of a mathematical theory of molecular structure. This is done through using comprehensive analytical tools namely the topological analysis and the Catastrophe theory (Bader 1990). II. The mechanical properties of AIM, the Ehrenfest forces operative on AIM, regional torques, regional energies, etc. are all introduced assuming the usual electronic Hamiltonian that is composed of electrons kinetic energy and the coulombic interactions of electrons and clamped nuclei as well as electronic interactions. Accordingly, in the regional hypervirial theorem, as the source of regional mechanical properties, the electronic Hamiltonian is employed throughout calculations (Bader 1990). Indeed, this framework itself is a wide context since the ab initio non-relativistic quantum chemistry aims to derive electronic wavefunction using various approximate well-known computational schemes: HF, DFT, MP2, CISD, CCSD, CCSD(T),… (Helgaker et al. 2000); the QTAIM formalism is able to deal with all these electronic wavefunctions. What if one decides to modify or change the electronic Hamiltonian? After all, the electronic Hamiltonian is just a starting point for more accurate descriptions of molecular systems. Indeed, this fact has been considered in the orthodox QTAIM through considering

123

S. Shahbazian

the re-derivation of AIM assuming the presence of molecule in the external electric, magnetic or electromagnetic fields by augmenting electronic Hamiltonian with extra terms (Bader 1990). Our primary investigations on the consequences of replacing point like clamped nuclei with finite size nuclei were also directly motivated by the same question; mathematically, the whole enterprise was just considering the consequences of modifying the electron-nuclear potential energy term in the electronic Hamiltonian on the formulation of the QTAIM (Nasertayoob and Shahbazian 2010). All in all, the whole studies demonstrate that indeed the orthodox QTAIM (tenets I and II) is robust against such modifications. Do all observations indicate the robustness of the orthodox QTAIM and its constructs? As will be shown in forthcoming subsections the answer is no. The main point is that the orthodox QTAIM is a single-component theory; it assumes that only electrons, as quantum particles, are of relevance for the AIM analysis and the other ingredients, i.e. nuclei, are not directly contributing to the form and properties of the AIM. Let’s consider this point in more detail. Nuclear dynamics: first alarm Nuclei in molecules are in eternal vibrations. In contrast to the static view originating from the hand-made molecular models, this zero-point jiggling demonstrates that no single nuclear geometry is attributable to a molecule but a set of most accessible geometries. In such a dynamic view, what is the role of nuclear dynamics in the orthodox QTAIM? This is a subtle issue. Let’s first briefly consider the standard apparatus used within the context of the QTAIM for this purpose. As emphasized previously, the basic formulation of the QTAIM is based on the usual electronic Hamiltonian, neglecting nuclear kinetic energy terms, thus nuclear dynamics is not explicitly taken into account. On the other hand, in the clamped nucleus approximation, founded on the well-known Born–Oppenhiemer methodology (Born and Oppenheimer 1927), the electronic wavefunctions originating from the electronic Hamiltonian are parametrically dependent on the relative positions of nuclei. Accordingly, the one-electron densities are also parametrically dependent on nuclear coordinates. This parametric dependence guarantees that upon the change of the relative position of nuclei, for instance from their reference values at the equilibrium geometry, the one-electron density also responds and varies (Wiberg et al. 1991). This response is quantified by the topological analysis of the one-electron densities at various nuclear geometries disclosing AIM and different possible molecular graphs that the collection of which is known as ‘‘structural diagram’’ summarizing the evolution of molecular structures defined within the context of the QTAIM (Bader 2010); these diagrams are rationalized by the mathematical apparatus of the Catastrophe theory and are reminiscent of the usual phase diagrams in thermodynamics (Bader 1990; Collard and Hall 1977; Nasertayoob and Shahbazian 2008). Thus, according to this brief sketch, one may claim that the orthodox QTAIM, albeit implicitly, takes the nuclear dynamics into account. But, is this the whole story? To have a comprehensive AIM theory one needs to not only discriminate the morphology of AIM, deriving topological atoms by discerning the inter-atomic surfaces (the boundaries between AIM) (Bader 1990), but also attribute regional properties to each atomic basin. Accordingly, the regional energy, the energy of each atomic basin, is a key property. To derive these energies within the context of the QTAIM, one needs to use the regional virial theorem; the balance dictated by this regional theorem between the kinetic energy of electrons and the virial of forces exerting on electrons is vital for the partitioning of the total energy of a molecule into atomic contributions (Bader 1990). Out of

123

Beyond the orthodox QTAIM

equilibrium geometries, complications arise since the forces operative on the clamped nuclei are not null and they are also contributing to the balance dictated by the virial theorem (Levine 2000). A ‘‘Rigorous’’ partitioning of this new term into atomic contributions is not trivial and does not seem to be yet accessible [for a technical discussion check in particular Todd Keith’s contribution (Matta and Boyd 2007)]. How one must deal with the energy of nuclear vibrations within the context of the QTAIM without a firm definition of the basin energies at non-equilibrium geometries? How is it possible to determine the contribution of zero-point vibrations to each topological atom? Even if one attributes regional energy to a topological atom at various nuclear geometries with a floppy definition of regional energy at non-equilibrium geometries, then how is it possible to extract the ‘‘vibrational’’ contribution? Do we need to introduce a ‘‘mean regional energy’’ using a statistical model that attributes a weight to the regional energies of a topological atom calculated at various geometries? One may even proceed to more advanced questions: How one must determine the importance of various used regional energies in this statistical paradigm? Can quantum vibrational amplitudes derived from nuclear (vibrational–rotational) wavefunction solve the problem? How it is possible to quantify the contribution of excited, instead of zero-point, vibrational modes in the regional energies? What about rotations, and concomitant deformations, in non-rigid molecules? After all, any type of the geometrical deformation of a molecule, regardless to its origin, makes certain non-equilibrium geometries more accessible… This long list of questions does not seem to have clear cut answers in the orthodox QTAIM. In a personal communication the late Richard once stated that the incorporation of nuclear dynamics into the framework of the QTAIM is the most important remaining task in completing the orthodox framework. Ironically, it was emerged that materializing this task needs abandoning the orthodox framework. Let’s have a digression from this story in subsequent subsection and then return to it again in a wider context. Exotic species: ‘‘AIM-less’’ systems? The current list of fundamental particles considered in particle physics is too long (Khanna 2009). Most of them are extremely unstable decaying into more stable particles in very short times, thus uninteresting for majority of chemists. However, because of technological advances in producing and detection techniques of unstable particles in the last four decades, there was (and is) a growing interest in considering molecules containing particles other than electrons and stable nuclei. Some members of a special family of fundamental particles namely the leptons, enclosing also electron, are of most importance; positron, the anti-particle of electron and muon, the heavier analogue of electron are of special interest in chemistry (Walker 1986). It is tempting to call the study of the structure and reactivity of molecules containing these particles, positronic and muonic systems, as leptonic chemistry. Let’s consider them briefly. Positron is virtually indistinguishable from an electron in all its properties except from the fact that it carries a positive charge instead of the negative charge of electron. Upon addition of a positron to a molecule, a new bound quantum state may emerge that persist usually for nano-seconds; an electron and positron as a matter/anti-matter pair finally annihilate producing Gamma rays (Jean et al. 2003). Since positron has a positive charge, its interaction with electrons is attractive whereas repulsive with nuclei. Accordingly, the real space distribution of positron in molecules is much different from electrons; the distribution is extremely diffuse usually ‘‘behind’’ the most electronegative atom to avoid positively charged nuclei and benefiting mostly from favorable electron–positron

123

S. Shahbazian

interaction (for a list of relevant references and a technical discussion see Goli and Shahbazian 2011a). One may claim that because of the diffuse nature of positronic distribution, it usually acts only as a perturbation, albeit not always minor, to the electronic structure of positron-free molecule. Muons act quite differently since they are almost two hundred times more massive than electrons. The properties of minus muon are similar to electron whereas positive muon, the anti-particle of minus muon, is similar to positron neglecting the mass difference in both cases. However, in contrast to positron that is intrinsically stable, both free and bound muons are unstable particles decomposing into electrons/positrons and neutrinos (another allusive member of the leptons) in micro-seconds (Nagamine 2003). Because of its larger mass, in replacing an electron with an minus muon in an atom, a large shrinkage in muon real space distribution takes place; roughly speaking, minus muon rotates in a much smaller orbit around nucleus than a typical orbit of an electron (Nagamine 2003). On the other hand, based on comprehensive experimental and theoretical investigations (Fleming et al. 2011a, b; Espinosa-Garcı´a 2008), one may conclude that a positive muon acts like a lighter isotope of hydrogen. The main difference with proton or deuteron is that it is more amenable to tunneling and related purely quantum effects because of its smaller mass (Alexander 2011). What about the presence of AIM in the positronic and muonic (positive muon) species? Certainly the usual electronic Hamiltonian is not appropriate to consider the internal quantum structure of these species; neglecting the kinetic energies of positrons or muons is not legitimate since they are light particles. One the other hand, if one aims to consider positron and muon as quantum particles, then, not only the Hamiltonian need to be modified but also the one-densities of positron and muon similar to one-electron densities naturally emerge. How do these densities contribute to the shape of the AIM? How does each type of quantum particles contribute to the basin properties? The mathematical framework of the orthodox QTAIM does not seem to be suitable for answering these and similar questions since within its context, as emphasized previously, only electrons are involved in outlining its two basic tenets. Intuitionally, based on previous discussions, it seems ‘‘reasonable’’ (but not inevitable!) to assume that AIM would be similar before and after the addition of positron to a molecule. After all, if positron only acts as a perturbation, why the whole underlying AIM structure of a molecule must suddenly disappears upon the addition of a positron? The same intuition also points to the possibility that upon the replacement of a positive muon instead of a proton in a molecule, the hydrogen basin persists; a lighter isotope of proton seems capable, at least in principle, to shape its own topological atom… ‘‘Back to the future’’: beyond clamped nucleus approximation The problems of treating the nuclear dynamics within the context of the QTAIM as well as deriving AIM structure of the exotic species have a common point: Both are pointing to a novel AIM analysis for dealing with multi-component systems. Let’s consider this point in more detail. The Born–Oppenhiemer (BO) ‘‘paradigm’’ invokes a well-known two-step procedure: First considering electrons’ dynamics through solving the electronic Schro¨dinger equation assuming clamped nuclei, and then considering nuclear dynamics through solving the nuclear Schro¨dinger equation taking the electronic energies (plus nuclear repulsions) as potential energy terms. This is an adiabatic view however it is always possible to include non-adiabatic (coupled) nuclei-electron dynamics in a third step (Baer 2006). In

123

Beyond the orthodox QTAIM

considering possible extension of the orthodox QTAIM including nuclear dynamics, this adiabatic point of view prompts to conceive that nuclear dynamics must be somehow added to the orthodox, only electron, AIM analysis and subsequently one may consider a non-adiabatic extension probably only as a minor modification; in the heart of the BO paradigm there is a time line: ‘‘yesterday’’, electrons plus clamped nuclei, ‘‘today’’, adiabatic nuclear dynamics, and ‘‘tomorrow’’, non-adiabatic nucleus-electron coupled dynamics. It seems to this author that the atmosphere imprinted by this well-established time line was the source of the mental resistance to see how a ‘‘back to the future’’ approach may solve the problem in a simple way (according to Popelier’s interesting metaphor, the time line acts as a ‘‘circle’’. Check his wonderful preface in Popelier 2011). In the ‘‘back to the future’’ approach one may start from the outset with quantum instead of clamped nuclei. In other words, no adiabatic separation is assumed in first place but electrons and nuclei are considered on equal footing as genuine quantum waves. This methodology is inherently a non-adiabatic view in nature however this ‘‘non-adiabaticity’’ is not viewed as a minor modification to a reference, predefined, adiabatic state. Fortunately, in the last 15 years, a number of research groups have been involved in developing efficient ab initio methodologies to solve Schro¨dinger equation simultaneously for both electrons and nuclei in polyatomic molecules (Cafiero et al. 2003; Nakai 2007; Ishimoto et al. 2009). This methodologies yield the raw material, proper wavefunctions, needed for an AIM analysis. The same computational methodologies are capable of performing ab initio calculations on the exotic species. Now, we are faced with a unified statement of two seemingly separate problems; by treating nuclei, positron and muon as quantum particles we deal with multi-component quantum systems. The multi-component nature emphasizes on the fact that in all such systems at least two quantum particles are involved in shaping the ‘‘quantum structure’’. Thus, the key question is now reduced to: How can one formulate an extended QTAIM for the multi-component systems? The answer to this question opens the door to a theory of molecular form, structure and AIM untied with clamped nuclei approximation thus going beyond the BO paradigm…

Current status: what we know and what remains to be known The previous discussions demonstrated the need as well as the general nature of an extended QTAIM; however, a concrete formulation was not an easy task since both mentioned tenets of the orthodox QTAIM must be modified simultaneously to accommodate the multi-component nature of the target systems. After many failed attempts and hundreds of pages of adverse calculations the consistent formulation was finally uncovered. In the following subsection only the history of the successful path, the ‘‘golden path’’, is narrated leaving the failed paths to freemason’s hall. Also, for simplicity, the twocomponent formulation is considered in this section leaving a brief sketch of the multicomponent version to the subsequent section. The Gamma… The first question one envisages regarding the extended QTAIM is: What is the scalar field to replace the one-electron density in the topological analysis? Since the morphology of AIM as well as the discrimination of basin boundaries emerge from the topological analysis, the answer to this question is pivotal for a successful extension. The first tentative

123

S. Shahbazian

clue emerged from seeking a mere formal analogy between the basic scalar fields of the one/multi-component DFT (Parr and Yang 1989) and that of the orthodox/extended QTAIM. Let’s show it in a simple scheme for the two-component systems:

DFT

QTAIM

One-electron density: qe ðqÞ One-densities and masses of electrons and the positively charged quantum particles: q ðqÞ; qþ ðqÞ; m ; mþ

, ?

,

One-electron density: qe ðqÞ Cðq ðqÞ; qþ ðqÞ; m ; mþ Þ

The second line in this scheme is based on an extended form of the Hohenberg–Kohn theorem first derived by Parr and coworkers (Capitani et al. 1982; Parr and Yang 1989); based on this theorem one only needs to know the one-densities of each type of quantum particles and their masses to be able to reconstruct the whole properties of two-component systems. While this is a step forward, mere analogy does not elucidates the explicit form of C, the Gamma. Based on a simple transformation between the extended Lagrangian and Hamiltonian forms of kinetic energy densities, both used also within the context of the orthodox QTAIM, it is quite straightforward to propose the explicit form of the Gamma (for technical details check section 3 in Nasertayoob et al. 2011):   m q ðqÞ ð1Þ CðqÞ ¼ q ðqÞ þ mþ þ Minus and plus subscripts are used to denote the one-density and the mass of electrons and positively charged particles (positrons, muons, protons, deuterons,…), respectively. Interestingly, this equation not only yields proper AIM as discussed below but also fits properly with other ingredients of the extended QTAIM (Heidar Zadeh and Shahbazian 2011) that in this stage of development deserves to be called the two-component QTAIM (TC-QTAIM). Before proceeding further, it is timely to emphasize that upon the increase of the mass of the positive particles, clamping the positive particles, the Gamma field approaches the one-electron density and in the infinite mass limit the correspondence is perfect as demonstrated very recently: limmþ !1 CðqÞ ! qe ðqÞ (Goli and Shahbazian 2012). This is an important result demonstrating the non-trivial fact that in the limit of infinite mass the TC-QTAIM recovers the orthodox QTAIM results; the former encompasses the latter as one expects from a consistent generalization of a methodology. ! The gradient vector field of the Gamma, r CðqÞ, reveals the critical points, ! ! r Cðq Þ ¼ 0, and the inter-atomic surfaces satisfying: r CðqÞ  ! n ¼ 0, thus delineating C

the ‘‘basins of attractions’’ each representing a topological atom (Goli and Shahbazian 2012). After establishing the morphology of atomic basins, one needs to determine the regional properties of AIM. Properties of AIM: the density combination strategy Equation (1) not only introduces the basic scalar field for the topological analysis, but also reveals a delicate feature of the TC-QTAIM, the rule of combining densities; in Eq. (1) a single variable, ‘‘united’’ variable, is used for both one-densities while these two densities refer to different types of quantum particles. The same rule is used to introduce combined

123

Beyond the orthodox QTAIM

property densities the integration of which yield the regional properties of the topological atoms; the combined property densities are constructed from the combination of two property densities originating separately from electrons/positively charged particles. For a property, say M, the basic formulation is summarized as follows: ~ MðqÞ ¼ M ðqÞ þ Mþ ðqÞ Z ~ ~ MðXÞ ¼ dq MðqÞ

ð2Þ

X

~ ~ MðXÞ is the regional property of an atomic basin, X, whereas MðqÞ, the combined property density, may be evaluated at every point in space as a local manifestation of the considered property. Based on the experiences within the context of the orthodox QTAIM, an interesting point for sampling properties is the bond critical points (BCP), (3, -1) critical points usually (but not always) between each pair of adjacent nuclei (Bader 1990); the resulting topological indexes, the amount of properties at the BCP point, are used for deciphering the nature of bonding (Goli and Shahbazian 2011a, b). Primary calculations demonstrate that in the case of heavy particles like protons and also its heavier isotopes, all described by localized quantum waves, the contribution of Mþ ðqÞ at the BCP is virtually null, Mþ ðqC Þ  0, and the combined density is equal to M ðqC Þ (Goli and Shahbazian 2011b). However, M ðqC Þ has the imprint of positive particle’s mass since even at the HF computational level, the electronic part of ab initio wavefunctions is obtained within the mean electric field of positive particles; the mean field of positive particles that ‘‘polarizes’’ the electronic distribution is determined by their charge density distribution which explicitly depends on the mass of the particle. Thus, M ðqC Þ is slightly different for various isotopes as well as systems with and without positron revealing subtle differences in their bonding ~ modes. Basin properties, MðXÞ, is also used to rationalize the delicate differences of the topological atoms corresponding to various isotopes as well as positron–molecule complexes as will be discussed in subsequent subsection. Similar to the Gamma field, one expects that upon increasing the mass of the positive ~ ! Me ðqÞ, particle all property densities satisfy the following relation: limmþ !1 MðqÞ where Me ðqÞ is the property density obtained within the context of the orthodox QTAIM (Goli and Shahbazian, under preparation). This relationship clearly demonstrates that not only the morphology of the orthodox AIM, but also their properties are recovered from the TC-QTAIM by clamping the positive particles attributing to each an infinite mass. How one must determine the explicit form of M ðqÞ? Within the context of the orthodox QTAIM the density of most properties, particularly mechanical properties, are introduced by the regional hypervirial theorem (Bader 1990); a regional version of the putative hypervirial theorem for the whole molecule (Levine 2000). The idea of density combination may be also used to derive an extended regional hypervirial theorem (for a technical discussion check section 3 in Goli and Shahbazian 2012). Although in this contribution it is hesitated to go into technical details, it is illustrative to state the twocomponent local virial theorem:  2  h T ~ ~ r2 CðqÞ ð3Þ 2TðqÞ ¼ V ðqÞ þ 4m ~ In this equation TðqÞ and V~T ðqÞ stand for the combined kinetic energy density and the combined density of the total virial of forces, respectively (Goli and Shahbazian 2012). For a topological atom based on Gauss’s theorem (Nasertayoob et al. 2011):

123

S. Shahbazian

H ! dqr2 CðqÞ ¼ oX dS r CðqÞ  ! n , the regional integration yields the two-component regional virial theorem that is used to introduce the regional energy of an atomic basin:

R

X

~ 2TðXÞ ¼ V~T ðXÞ ~ E~ðXÞ ¼ TðXÞ þ V~T ðXÞ ¼ T~ðXÞ ¼ T ðXÞ  Tþ ðXÞ

ð4Þ

This is an interesting result since it demonstrates that the energy of each atomic basin originates from the regional kinetic energies of both electrons and positively charged particles. In the case of heavy particles, Tþ ðXÞ is dominated by vibrational kinetic energy thus directly incorporating the nuclear vibrational dynamics in the regional energies (for some numerical comparisons of Tþ ðXÞ with the zero-point energies, derived within the adiabatic scheme, check subsection 3.2 in Goli and Shahbazian 2011b). Lithium hydride as a benchmark The computational TC-QTAIM studies were conducted on some diatomic positronic species as well as lithium hydride and its isotopomers (Goli and Shahbazian 2011a, b); in all studies reasonable AIM morphologies and properties were observed. In this subsection only the results on LiH and its positronic complex, LiH,e?, as typical examples are briefly reviewed. Figure 1a depicts the hydrogen atomic basin of LiH,e?. Basin integration of positronic population demonstrates that virtually the whole population of positron is contained in the hydrogen basin. As also stated previously, this is a general observation in line with high level ab initio calculations on positron’s real space distribution (Strasburger 1999); positron has an extremely diffuse distribution in the hydrogen basin. The large regional positive electric dipole of the hydrogen basin conforms well with these facts as well as the observation that the positive contribution to regional energies, Tþ ðXÞ, is non-zero only for the hydrogen basin. The latter fact was used to introduce the regional positron affinities as a novel concept (for relevant discussion check subsection 6.2 in Goli and Shahbazian 2011a). Although positron is not directly contributing in the topological indexes, having almost no probability of observation at the BCP between the two nuclei, it affects these indexes indirectly through changing, albeit slightly, the geometry and the electronic distribution of positron–molecule complexes relative to original positron-free molecule. Figure 1b depicts the hydrogen basin in LiH system. In this case there is no clamped hydrogen nucleus since it is treated as a quantum wave; roughly speaking, we are faced with an atomic basin containing only quantum waves. However, basin integration of proton population clearly demonstrates that the whole positive particle population is contained in the hydrogen basin as expected intuitionally. The positive contribution to the regional energy in this basin is comparable in magnitude with the zero-point energy calculated for the clamped nucleus model of LiH. Probably, the most interesting observation is the variations of the morphology and properties of this basin upon replacing proton with deuteron and tritium. The positive contribution of the regional energy of the hydrogen basin diminishes with the increase of the nuclear mass inline with the reduction of zeropoint energies from LiH to LiD and then LiT. The morphology of the corresponding basins also varied slightly upon the variation of nuclear mass and the resulting population of electrons increases by increasing the nuclear mass demonstrating the parallel increase of electronegativity upon the increase of the nucleus mass (for a discussion on other studies confirming this trend check subsection 3.2 in Goli and Shahbazian 2011b). These examples

123

Beyond the orthodox QTAIM

Fig. 1 a The hydrogen basin in positron-lithium hydride complex. The red dot is the bond critical point that is linked to the gray sphere, lithium nucleus, by a bond path. b The hydrogen basin in lithium hydride without a clamped hydrogen nucleus. The violet dots are global attractors, (3, -3) critical points, whereas the red dot is a bond critical point. The global attractor out of hydrogen basin is located very near to the clamped lithium nucleus. In both panels C = 0.01 au iso-surface is used to depict the outer boundary of the atomic basins. (Color figure online)

demonstrate that the TC-QTAIM is able to distinguish the topological atoms associated to various isotopes of an element. All in all, the TC-QTAIM is capable not only incorporating nuclear vibrational dynamics into the regional properties, but also yielding numerical and conceptual results for an AIM analysis of the exotic species. The computational results are non-trivial and in many cases comparison with the associated results of the orthodox QTAIM (on the clamped nucleus systems) demonstrates that basin properties are seriously affected beyond the usual ‘‘chemical accuracy’’ by the presence of positron or nuclear vibrational dynamics; the results of the TC-QTAIM are not just minor (negligible) corrections to the orthodox QTAIM analysis (Goli and Shahbazian 2012).

123

S. Shahbazian

What we need to know: localization/delocalization indexes and response properties The formalism developed in previous contribution though demonstrates the feasibility of an extended QTAIM, still is missing two main ingredients; the localization/delocalization measures as well as the response properties of AIM like polarizability and magnetizability are part of the orthodox QTAIM (Bader 1990). The localization/delocalization measures (indexes) are important tools in the TC-QTAIM analysis since, even intuitionally, one expects that for light particles as well as electrons quantum delocalization is an important effect. Our very recent theoretical investigations indeed points to the fact that the extended localization/delocalization indexes for positive particles must be mass dependent; computational studies on a set of four body systems composed of two electrons and two positively charged particles with a variable mass reveals unambiguously this mass dependent pattern (Goli and Shahbazian, under preparation). As emphasized previously, in the presence of external electric or magnetic fields the basic structure of AIM persists though the morphology and properties of AIM are perturbed according to the strength and nature of the fields. Within the QTAIM methodology, the response of molecules to the week external fields is quantified introducing basin polarizabilities and magnetizabilities; this is a regional view considering the share of each topological atom in the total response of a molecule to an external field. In order to evaluate the role of nuclear dynamics in these response properties, one needs to introduce the other quantum particles’ contribution separately from electrons’ contribution and then applying the rule of combination of densities (Shahbazian, under preparation). The resulting formulation set the stage for the computational TC-QTAIM analysis of systems containing quantum nuclei and exotic species within various external fields.

Prospects and challenges The ‘‘road map’’ disclosed in previous section opens novel paths to tackle problems/ questions that have not yet found their proper solutions/answers. In this section, some of these problems are stated very briefly leaving comprehensive treatments for future contributions. This section by its very nature is somehow speculative though, our yet unpublished investigations points that we are just in the edge of materializing proper solutions. There are also wilder speculations, like the possible applications of the TC-QTAIM formalism in nuclear structure theory as well as in the superconductivity and also its extension for relativistic phenomena… Extending further: multi-component version Probably, the most important remaining theoretical task is the extension of the TC-QTAIM into the multi-component version, MC-QTAIM. It is possible to demonstrate that in this extension the Gamma field is replaced with following scalar field: CðPÞ ðqÞ ¼ q1 ðqÞ þ

P X

ðm1 =mn Þqn ðqÞ

ð5Þ

n¼2

In this equation, the subscript P is used to denote the ‘‘type of components’’, quantum particles apart from the ubiquitous electrons that are labeled ‘‘1’’, used to construct the multi-component system under study; it is evident that this new scalar field is a linear

123

Beyond the orthodox QTAIM

extension of the Gamma field in Eq. (1). Therefore, the whole topological analysis within the context of the MC-QTAIM is performed on this novel field yielding AIM. A linear extension also permits attributing regional properties to the topological atoms: ~ MðqÞ ¼ M1 ðqÞ þ ~ MðXÞ ¼

Z

P X

Mn ðqÞ

n¼2

ð6Þ

~ dq MðqÞ

X

The extended multi-component regional hypervirial theorem is used to derive the explicit forms of the regional properties (Goli and Shahbazian, under preparation). It is tempting to call P in Eqs. (5) and (6) as the cardinal number of the extended theory; putting P = 1 the ‘‘single-component’’ orthodox QTAIM results whereas P = 2 yields the TC-QTAIM. All developed or in progress theoretical ingredients of the TC-QTAIM must be also extended within the context of the MC-QTAIM… Finite temperatures: crystals and other extended systems According to the best of author’s knowledge all AIM methodologies have been formulated as ‘‘zero-temperature’’ theories. This is reasonable for free molecules but in extended systems, e. g. crystals, at room temperature various vibrational modes are populated at thermal equilibrium; various ground and vibrationally excited modes coexist simultaneously. Accordingly, the corresponding nuclear vibrational dynamics is complex and even the completed MC-QTAIM is unable to deal with such situations. Thus, a finite-temperature AIM methodology is needed for a comprehensive AIM analysis of crystals. The development of such methodology is timely taking the fact that interest in the orthodox QTAIM is growing rapidly in crystallographers’ community. A density matrix based formalism is now under consideration in our group and the primary results seem to be promising (Shahbazian, under preparation). Beyond molecular-fixed frames: touching the ‘‘molecular structure conundrum’’ So far in all computational TC-QTAIM studies the used wavefunctions have been derived assuming a molecular-fixed frame that rotates with molecule; thus, the nuclear dynamics is confined to vibrational dynamics. As first noted by Woolley and then Sutcliffe, a more rigorous quantum treatment is based on a frame that is translationally invariant but contains the rotational dynamics (Woolley 1976, 1978, 1985, 1986; Sutcliffe and Woolley 2005a). The emerging question is that how one may derive the ‘‘internal structure’’ of a molecule (call it geometry, shape, molecular graph, AIM,…) from a wavefunction that contains the rotational invariance. This is not the place to go into subtle technical details but it suffice to emphasize that in contrast to some developments (Cafiero and Adamowicz 2004, 2007; Ma´tyus et al. 2011; Ma´tyus and Reiher 2011) ‘‘structure extraction’’ has not yet been fully addressed and is named judiciously by Weininger as molecular structure conundrum (Weininger 1984). Taking the recent advances in the ab initio non-BO calculations (Cafiero et al. 2003) and the accessibility to proper wavefunctions, tackling this riddle seems to be an urgent need; in a recent comment Sutcliffe and Woolley have colorfully restated various aspects of this riddle in an accessible non-technical language (Sutcliffe and Woolley 2005b). From a more practical point of view, one may point to the fact that in non-rigid molecules rotational dynamics is

123

S. Shahbazian

not irrelevant since it directly affects other internal motions; rotations are not completely separable from other ingredients of internal dynamics (Sutcliffe 2000). Then, how does AIM emerge from rotationally invariant multi-component wavefunctions? Our recent, yet unpublished, investigations demonstrate that one needs to modify the TC(MC)-QTAIM formalism in order to ‘‘see’’ the AIM structure in such wavefunctions (Goli and Shahbazian, under preparation). Thus the story of full incorporation of nuclear dynamics within the context of AIM methodology is far from complete… ‘‘Seeing’’ the quantum tunneling The considered systems in all previous computational studies were not suspect for quantum tunneling however there are many interesting molecules where quantum tunneling actively participates in shaping their structure and reactivity; Malonaldehyde is a typical example (Fillaux and Nicolaı¨ 2005). It is indeed interesting to consider the AIM, as a real space structure, of malonaldehyde and similar systems to see how proton tunneling manifest itself in the TC(MC)-QTAIM analysis… The realm of Schro¨dinger’s cat: the art of black magic The case of proton tunneling is also interesting since one is faced with a quantum superposition of two or more ‘‘classical molecular structures’’ (Fillaux and Nicolaı¨ 2005); this is also part of the molecular structure conundrum (Sutcliffe and Woolley 2005b; Primas 1982). What would be the result of the TC(MC)-QTAIM analysis of a wavefunction that describes a superposition of two classical structures? This is the realm of the ‘‘Alice’’, the art of black magic… Toward extremes: hydrogen under pressure Since protons are much more massive than electrons, the quantum wave attributed to a proton is much more localized than the waves of electrons. This localized nature prevents in most cases, apart from systems with quantum tunneling, an efficient overlap of protons wavefunctions and concomitant delocalization. However, under extreme pressures, the protons must stay ‘‘near’’ and ‘‘touch’’ each other thus the overlap of nuclear quantum waves is inevitable; in such cases the large mass of proton does not prevent quantum delocalization (Goncharov and Crowhurst 2007). Taking that fact that hydrogen is transferred into a novel metallic phase at high pressures (Mao and Hemley 1994; Eremets and Troyan 2011; Klug and Yao 2011), it is interesting to consider the role of proton delocalization in this phase transition within the context of the TC-QTAIM or a finite temperature AIM methodology. Accordingly, the localization/delocalization measures of the TC-QTAIM may reveal strange characteristics for the atoms in ultra-dense hydrogen… The final words The considered examples and the list of possible projects in the previous sections all clearly reveal the fact that AIM methodology is a research program rather than a closed theory; the orthodox QTAIM is just the first step in ‘‘Jacobs’s ladder’’. The TC-QTAIM is the second step and the MC-QTAIM is the third, subsequent steps are also revealing themselves…

123

Beyond the orthodox QTAIM

As the final words let me also ask the final question: Is the orthodox QTAIM just the tip of an iceberg? If the reader of this paper is justified that the answer is ‘‘yes’’ then it is safe to claim that Richard’s legacy is alive, promising new wonderlands… Acknowledgments The author is grateful to Dr. Cherif Matta, Dr. Paul Ayers, Miss. Farnaz Heidar Zadeh, Dr. Cina Foroutan-Nejad and Mr. Mohammad Goli for detailed reading of a previous draft and their fruitful comments and suggestions. The author is also indebted to his current Ph.D student Mohammad Goli for his dedicated work and outstanding contributions.

References Alexander, M.H.: Chemical kinetics under test. Science 331, 411–412 (2011) Bader, R.F.W.: Atoms in Molecules: A Quantum Theory. Oxford University Press, Oxford (1990) Bader, R.F.W.: Definition of molecular structure: by choice or by appeal to observation? J. Phys. Chem. A 114, 7431–7444 (2010) Baer, M.: Beyond Born–Oppenhiemer. Wiley, New Jersey (2006) Born, M., Oppenheimer, R.: Zur Quantentheorie der Molekeln. Ann. Phys. 84, 457–484 (1927) Cafiero, M., Adamowicz, L.: Non-Born–Oppenhiemer calculations of the ground state of H3. Int. J. Quantum Chem. 107, 2679–2686 (2007) Cafiero, M., Adamowicz, L.: Molecular structure in non-Born–Oppenhiemer quantum mechanics. Chem. Phys. Lett. 387, 136–141 (2004) Cafiero, M., Bubin, S., Adamowicz, L.: Non-Born–Oppenhiemer calculations of atoms and molecules. Phys. Chem. Chem. Phys. 5, 1491–1501 (2003) Capitani, J.F., Nalewajski, R.F., Parr, R.G.: Non-Born–Oppenhiemer density functional theory of molecular systems. J. Chem. Phy. 76, 568–573 (1982) Collard, K., Hall, G.G.: Orthogonal trajectories of the electron density. Int. J. Quantum Chem. 12, 623–637 (1977) Eremets, M.I., Troyan, I.A.: Conductive dense hydrogen. Nat. Mater. 10, 927–931 (2011) Espinosa-Garcı´a, J.: Theoretical rate constants and kinetic isotope effects in the reaction of the methane with H, D, T, and Mu atoms. Phys. Chem. Chem. Phys. 10, 1277–1284 (2008) Fillaux, F., Nicolaı¨, B.: Proton transfer in malonaldehyde: from reaction path to Schro¨dinger’s Cat. Chem. Phys. Lett. 415, 357–361 (2005) Fleming, D.G., Arseneau, D.J., Sukhorukov, O., Brewer, J.H., Mielke, S.L., Schatz, G.C., Garrett, B.C., Peterson, K.A., Truhlar, D.G.: Kinetic isotope effects for the reactions of muonic helium and muonium with H2. Science 331, 448–450 (2011a) Fleming, D.G., Arseneau, D.J., Sukhorukov, O., Brewer, J.H., Mielke, S.L., Truhlar, D.G., Schatz, G.C., Garrett, B.C., Peterson, K.A.: Kinetics of the reaction of the heaviest hydrogen atom with H2, the 4 Hel ? H2 ? 4HelH ? H reaction: experiments, accurate quantal calculations, and variational transition state theory, including kinetic isotope effects for a factor of 36.1 in isotopic mass. J. Chem. Phys. 331, 448–450 (2011b) Gatti, C., Macchi, P.: Modern Charge-Density Analysis. Springer, New York (2012) Goli, M., Shahbazian, S.: The quantum theory of atoms in positronic molecules: a case study on diatomic species. Int. J. Quantum Chem. 111, 1982–1998 (2011a) Goli, M., Shahbazian, S.: Atoms in molecules: beyond Born–Oppenheimer paradigm. Theor. Chem. Acc. 129, 235–245 (2011b) Goli, M., Shahbazian, S.: The two-component quantum theory of atoms in molecules (TC-QTAIM): foundations. Theor. Chem. Acc. 131(1–19), 1208 (2012) Goncharov, A.F., Crowhurst, J.: Proton delocalization under extreme condition of high pressure and temperature. Phase Transit. 80, 1051–1072 (2007) Helgaker, T., Jørgenson, P., Olsen, J.: Molecular Electronic-Structure Theory. Wiley, New York (2000) Heidar Zadeh, F., Shahbazian, S.: The quantum theory of atoms in positronic molecules: the subsystem variational procedure. Int. J. Quantum Chem. 111, 1999–2013 (2011) Ishimoto, T., Tachikawa, M., Nagashima, U.: Review of multicomponent molecular orbital method for direct treatment of nuclear quantum effect. Int. J. Quantum Chem. 109, 2677–2694 (2009) Jean, Y.C., Mallon, P.E., Schrader, D.M.: Principles and Applications of Positron & Positronium Chemistry. World Scientific, New Jersey (2003) Khanna, M.P.: Introduction to Particle Physics. PHI Learning, New Delhi (2009)

123

S. Shahbazian Klug, D.D., Yao, Y.: Metallization of solid hydrogen: the challenge and possible solutions. Phys. Chem. Chem. Phys. 13, 16999–17006 (2011) Levine, I.N.: Quantum Chemistry. Prentice-Hall, New Delhi (2000) Mao, H.-K., Hemley, R.J.: Ultrahigh-pressure transitions in solid hydrogen. Rev. Mod. Phys. 66, 671–692 (1994) Ma´tyus, E., Hutter, J., Mu¨ller-Herold, U., Reiher, M.: On the emergence of molecular structure. Phys. Rev. A 83, 052512 (2011) Ma´tyus, E., Reiher, M.: Extracting elements of molecular structure from the all-particle wave function. J. Chem. Phys. 135, 204302 (2011) Matta, C.F., Boyd, R.J.: The Quantum Theory of Atoms in Molecules: From Solid State to DNA and Drug Design. Wiley, Weinheim (2007) Nagamine, K.: Introductory Muon Science. Cambridge University Press, Cambridge (2003) Nakai, H.: Nuclear orbital plus moeluclar orbital theory: simultaneous determination of nuclear and electronic wave functions without Born-Oppenheimer approximation. Int. J. Quantum Chem. 107, 2849–2869 (2007) Nasertayoob, P., Goli, M., Shahbazian, S.: Toward a regional quantum description of the positronic systems: primary considerations. Int. J. Quantum Chem. 111, 1970–1981 (2011) Nasertayoob, P., Shahbazian, S.: The topological analysis of electronic charge densities: a reassessment of foundations. J. Mol. Struct. (Thoechem) 869, 53–58 (2008) Nasertayoob, P., Shahbazian, S.: Revisiting the foundations of the quantum theory of atoms in molecules (QTAIM): the subsystem variational procedure and the finite nuclear models. Int. J. Quantum Chem. 110, 1188–1196 (2010) Parr, R.G., Yang, W.: Density-Functional Theory of Atoms and Molecules. Oxford University Press, Oxford (1989) Popelier, P.: Atoms in Molecules: An Introduction. Prentice Hall, London (2000) Popelier, P.: Solving The Schro¨dinger Equation: Has Everything Been Tried? Imperial College Press, London (2011) Primas H. (1982) Chemistry, quantum mechanics and reductionism. In: Lecture Notes in Chemistry, vol. 24. Springer, Berlin Shahbazian, S., Zahedi, M.: The role of observables and non-observables in chemistry: a critique to chemical language. Found. Chem. 8, 37–52 (2006) Shahbazian, S., Zahedi, M.: Letter to the editor: the concept of chemical bond—some like it fuzzy but others concrete. Found. Chem. 9, 85–95 (2007) Strasburger, K.: Binding energy, structure, and annihilation properties of the positron-LiH molecules complex, studied with explicitly correlated Gaussian functions. J. Chem. Phys. 111, 10555–10558 (1999) Sutcliffe, B.T.: The decoupling of electronic and nuclear motions in the isolated molecular Schro¨dinger Hamiltonian. Adv. Chem. Phys. 114, 1–121 (2000) Sutcliffe, B.T., Woolley, R.G.: Molecular structure calculations with clamping the nuclei. Phys. Chem. Chem. Phys. 7, 3664–3676 (2005a) Sutcliffe, B.T., Woolley, R.G.: Comment on ‘‘Molecular structure in non-Born–Oppenhiemer quantum mechanics’’. Chem. Phys. Lett. 408, 445–447 (2005b) Walker, D.C.: Leptons in chemistry. Acc. Chem. Res. 18, 167–173 (1986) Weininger, S.J.: The molecular structure conundrum: can classical chemistry be reduced to quantum chemistry. J. Chem. Educ. 61, 939–944 (1984) Wiberg, K.B., Hadad, C.M., Breneman, C.M., Laidig, K.E., Murcko, M.A., LePage, T.J.: The responses of electrons to structural changes. Science 252, 1266–1272 (1991) Woolley, R.G.: Quantum theory and molecular structure. Adv. Phys. 25, 27–52 (1976) Woolley, R.G.: Must a molecule have a shape? J. Am. Chem. Soc. 100, 1073–1078 (1978) Woolley, R.G.: The molecular structure conundrum. J. Chem. Educ. 62, 1082–1084 (1985) Woolley, R.G.: Molecular shapes and molecular structures. Chem. Phys. Lett. 125, 200–205 (1986)

123