The quantum theory of atoms in positronic molecules: The subsystem variational procedure

This contribution deals with the subsystem variational procedure within the context of the quantum theory of atoms in positronic molecules (QTAIPM). Before introducing the subsystem energy functional termed as joint subsystem energy functional, a novel notation and the combination strategy are disclosed in detail by restating the positronic subsystem hypervirial theorem. They are employed in proposing the proper subsystem energy functional, the validity of which is checked by various criteria. The zero flux surfaces of the joint density distribution are used to define the topological atoms in the positronic molecules, and they are incorporated into the subsystem variational procedure as proper real space boundary conditions. The variational procedure finally yields the flux of the joint current property density that also appears in the positronic subsystem hypervirial theorem. At every stage, the corresponding equations for the purely electronic systems within the context of the quantum theory of atoms in molecules (QTAIM) are presented to clearly reveal the analogy between these two formalisms and to emphasize the importance of combining the property density distributions in the QTAIPM. The presented material demonstrates the internal consistency of the whole framework and discloses the fact that the QTAIM must be regarded as a variant of the QTAIPM. Furthermore, this formalism promises an extended QTAIM, which is hoped to resolve the issue of molecular structure beyond the clamp nuclei approximation. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Atoms in molecules: beyond Born–Oppenheimer paradigm

This contribution presents the first atoms in molecules study that goes beyond the Born–Oppenheimer paradigm employing the newly developed two-component quantum theory of atoms in molecules (TC-QTAIM). The LiH, LiD, and LiT systems containing quantum instead of clamped hydrogen nuclei are used as typical examples. The computational analysis that is done on non-adiabatic wavefunctions derived from the fully variational multicomponent molecular orbital approach (FV-MC-MO) results in hydrogen atomic basins without any clamped nucleus. The topological analysis of the Γ-field, the field that replaces the usual one-electron density used in the orthodox topological analysis, reveals delicate differences among the considered systems. The calculation of basin properties also demonstrates that the TC-QTAIM differentiates among atomic basins containing isotopes. Since the nuclear dynamics is contained intrinsically in non-adiabatic wavefunctions, the nuclear contribution to both topological analysis and basin properties naturally emerges from the TC-QTAIM analysis resolving the long-standing obstacle of consistent incorporation of nuclear dynamics within the context of the orthodox QTAIM. Also, a similar analysis is done on non-adiabatic wavefunctions describing excited instead of ground nuclear vibrations of the considered systems demonstrating the fact that TC-QTAIM is capable of being employed for both ground and excited nuclear vibrational states.     

The two-component quantum theory of atoms in molecules (TC-QTAIM): foundations

The foundations of the two-component quantum theory of atoms in molecules (TC-QTAIM) are addressed in this contribution. In this regard, the theory is presented in an axiomatic manner and the main theorems describing regional properties of atoms in molecules are considered in detail. This is an extension of the orthodox quantum theory of atoms in molecules (QTAIM) for dealing with non-adiabatic wavefunctions of usual molecules as well as extracting the regional quantum structure of exotic species from the corresponding wavefunctions. The best examples of the latter are positronic and muonic species. The computational study of a model system consisting of a clamped lithium nucleus, four electrons, and a positively charged quantum particle carrying a unit of positive charge with a variable mass, m = 200–10¹³ m _e, supplements the theoretical argument demonstrating unambiguously that the TC-QTAIM analysis yields reasonable results. It reveals that the contribution of the positively charged particle in the topological analysis and basin properties is non-negligible. Most importantly, it is demonstrated that by increasing the mass of the positive particle, the TC-QTAIM analysis tends toward the QTAIM analysis of the lithium hydride system considered within the clamped nucleus paradigm. This result seems to indicate that the orthodox QTAIM is just the asymptote of the TC-QTAIM, the latter encompasses the former. Thus, one may claim that the TC-QTAIM is a unified framework for the AIM analysis of vast variety of quantum systems.     

Revisiting the foundations of the quantum theory of atoms in molecules: Toward a rigorous definition of topological atoms

An explicit classification of consistent variational constraints within the context of the “quantum theory of proper open subsystems” as well as the “quantum theory of atoms in molecules” (QTAIM) it presented. It is demonstrated that the general variational procedure is not sensitive enough to discriminate between different mathematically consistent variational conditions. The uniqueness of the regional kinetic energy is employed to derive the net zero‐flux condition and the regions satisfying this condition are named as quantum divided basins. A modified form of the local zero‐flux is proposed in order to define topological atoms within the context of the orthodox QTAIM. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Toward a fuzzy atom view within the context of the quantum theory of atoms in molecules: quasi-atoms

The notion of quasi-atoms is introduced within the context of the quantum theory of atoms in molecules. Being a subset of the quantum divided basins that were introduced previously, quasi-atoms are the quantum subsystems that are practically indistinguishable from the topological atoms; thus, revealing the continuous evolution of quantum divided basins into topological atoms. This indistinguishability is rooted in the limited accuracy of chemical observations; they are not sensitive to discriminate a topological atom from its associated quasi-atoms. In this regard, it is disclosed that the set of quantum atoms is in a wide-range including members other than topological atoms; the quasi-atoms are concrete examples. Finally, the idea of the fuzzy set of atoms that is foreign to the disjoint partitioning schemes for which the orthodox QTAIM is a classic example is extended employing the set of quasi-atoms.     

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

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.     

What does shape a topological atom?

In this pedagogical communication after demonstrating the legitimacy for using the quantum theory of atoms in molecules (QTAIM) to non-Coulombic systems, Hookean H₂ ⁺/H₃ ²⁺ species are used for AIM analysis. In these systems, in contrast to their Coulombic counterparts, electron density is atom-like and instead of expected two/three topological atoms, just a single topological atom emerges. This observation is used to demonstrate that what is really “seen” by the topological analysis of electron densities is the clustering of electrons. The very trait of monotonic decay of electron density around the “centers” of clustering guarantees the appearance of topological atoms as basin of attraction of the gradient vector field of the electron density. Although observations with Hookean molecules may seem disappointing at first glance, a careful reasoning points to the fact that the QTAIM methodology is extendable to novel domains, by a knowledge of the morphology of underlying densities, beyond the typical Coulombic systems.     

On the Nature of the Positronic Bond

Recently it has been proposed that the positron, the anti-particle analog of the electron, is capable of forming an anti-matter bond in a composite system consists of two hydride anions and a positron [Angew. Chem. Int. Ed. 57 , 8859–8864 (2018)]. In order to dig into the nature of this novel bond the newly developed multi-component quantum theory of atoms in molecules (MC-QTAIM) is applied to this positronic system. The topological analysis reveals that this species is composed of two atoms in molecules, each containing a proton and half of the electronic and the positronic populations. Further analysis elucidates that the electron exchange phenomenon is virtually non-existent between the two atoms and no electronic covalent bond is conceivable in between. On the other hand, it is demonstrated that the positron density enclosed in each atom is capable of stabilizing interactions with the electron density of the neighboring atom. This electrostatic interaction suffices to make the whole system bonded against all dissociation channels. Thus, the positron indeed acts like an anti-matter glue between the two atoms.     

The quantum divided basins: A new class of quantum subsystems

The rigorous theory of the quantum divided basins (QDB), the quantum subsystems emerging from the net zero‐flux equation, is considered in this article. This framework, the quantum theory of proper open subsystems, is derived from the extension of the quantum theory of atoms in molecules to encompass the new class of quantum subsystems. It is demonstrated that the regional hypervirial theorem and the associated regional observables as well as the subsystem variational procedure are all expressible for the QDB. The history of QDB is briefly reviewed and the bundles, which were introduced by other researchers, are offered as typical examples whereas new examples of QDB (not yet mentioned in literature) are also presented. Based on some model systems as well as the nitrogen molecule, the regional properties and the morphologies of typical QDB are scrutinized. It is also demonstrated that the QDB may be used to study the fine structure of the electron localization and delocalization. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Revisiting the foundations of the quantum theory of atoms in molecules: The subsystem variational procedure and the finite nuclear models

The role of finite nuclear models (FNMs) is scrutinized within the context of the quantum theory of atoms in molecules (QTAIMs). It is demonstrated that the newly proposed analytic‐algebraic definition of the topological atoms is consistently extendable to the cases where a FNM is employed to construct the molecular hamiltonian. The whole variational procedure is reconsidered, and the insensitivity of final results relative to the employed FNMs is explicitly demonstrated. The analysis once again clearly demonstrates that the analytic‐algebraic condition is an independent axiom that must be added to the subsystem variational procedure to construct the QTAIMs. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Extending the Domain‐Averaged Exchange‐Correlation Energies Within the Context of the MC‐QTAIM: Tracing Subtle Variations Induced by Isotope Substitution

Recently, it has been demonstrated that the domain‐averaged exchange‐correlation energies, V^xc, are capable of tracing the covalent character of atom–atom interactions unequivocally and thus pave the way for detailed bonding analysis within the context of the quantum theory of atoms in molecules (QTAIM) [M. García‐Revilla, E. Francisco, P. L. Popelier, A. Martín Pendás, ChemPhysChem 2013 , 14, 1211–1218]. Herein, the concept of V^xc is extended within the context of the newly developed multicomponent QTAIM (MC‐QTAIM). The extended version, , is capable of analyzing nonadiabatic wavefunctions and thus is sensitive to the mass of nuclei and can trace “locally” the subtle electronic variations induced by isotope substitution. To demonstrate this capability in practice, ab initio nonadiabatic wavefunctions for three isotopically substituted hydrogen cyanide molecules, in which the hydrogen nucleus was assumed to be a proton, deuterium, or tritium, were derived. The resulting wavefunctions were then used to compute and it emerged that for the hydrogen–carbon bond, the was distinct for each isotopic composition and varied in line with chemical expectations. Indeed, the introduction of paves the way for the investigation of vast numbers of structural and kinetic isotope effects within the context of the MC‐QTAIM.     

Latin American contributions to quantum chemical topology

We survey the contributions from Latin American theoretical chemists to the field of quantum chemical topology (QCT) over nearly the last 30 years with emphasis on the developments and applications of the quantum theory of atoms in molecules (QTAIM). Applications of QCT in the fields of excited states, electron delocalization, chemical bond, aromaticity, conformational analysis, spectroscopic properties, and chemical reactivity are presented. We also consider the coupling of QTAIM with time-dependent density functional theory, the virial theorem in the Kohn-Sham method and the inclusion of electron dynamical correlation in the interacting quantum atoms method using coupled cluster and multi-configurational densities. Additionally, we describe the development of efficient algorithms for the calculation of topological properties derived from the electron density. This review is aimed not only at providing an account of the contributions to QCT in Latin America but also at stimulating guides for further progress in the field.     

Hydrogen bond in the H3O(Ph3PO) 3+ complex. Features of the electron density distribution

A quantum chemical study of the hydrogen bond in the H₃O(Ph₃PO)₃ ⁺ complex is carried out using topological methods: quantum theory of atoms in molecules (QTAIM) and of electron localization function (ELF) theory. In the H₃O(Ph₃PO)₃ ⁺ complex, three lone electron pairs on the oxygen atom in Ph₃PO are found to be combined in one ELF basin, and they all participate in the hydrogen bond formation. The observed topological features of the H₃O(Ph₃PO)₃ ⁺ complex are compared to the topological features of related complexes and the literature data.     

Interatomic exchange-correlation interaction energy from a measure of quantum theory of atoms in molecules topological bonding: A diatomic case

The potential relations between the measure of topological interatomic bonding—integrals of electron density with respect to internuclear axis over the corresponding quantum theory of atoms in molecules (QTAIM)-defined interatomic surface (IAS)—and interatomic exchange-correlation contributions from the interacting quantum atoms approach are discussed. The quantum chemical computations of 38 equilibrium diatomic systems at different levels of theory (HF, MP2, MP4SDQ, and CCSD) are invoked to support abstract considerations. Parameters of excellent correlations between IAS integrals and interatomic exchange-correlation energy are found by the optimization. The performance of these trends depends on the accuracy of the electronic correlation treatment. The resulting trends are a unique feature of equilibrium states, whereas more complicated dependencies are explored for several systems at non-equilibrium conditions. The relations of established trends with other IAS-based estimations of strength of bonding interactions between topological atoms and issues explored for multiatomic systems are briefly discussed.     

Spanning set of silica cluster isomer topologies from QTAIM

Structural and chemical properties of the building block of silica nanowires, (SiO(2))(6), are investigated with the theory of atoms and molecules (QTAIM). Twenty-five conformers have been analyzed, ten of which have not been reported before. We extend the silica (SiO(2))(6) topology phase space using QTAIM; the Poincaré-Hopf topological sum rules are applied and used to identify the spanning set of topologies, and this includes finding eight new distinct topologies that satisfy the Poincaré-Hopf relation. A simple phase diagram of the solutions of the Poincaré-Hopf relation is created with the aid of a new classification scheme to determine the boundary between topological stability and instability. Sum rules are then found to be applicable to any set of isomers. We determine that O-O bonding interactions exist for the silica (SiO(2))(6) conformers in regions where the energy surface is flattest. In addition, we identify unstable local minima in the topology of the charge density in order to further compare conformer instabilities. We quantify the dimensionality of a molecule using the Poincaré-Hopf relation instead of Euclidean geometry. This quantum topological definition of geometry shows that the four most energetically stable (SiO(2))(6) conformers are quantified as two-dimensional within the new quantum topology.     

Can QTAIM topological parameters be a measure of hydrogen bonding strength?

The block-localized wave function (BLW) method, which is the simplest variant of ab initio valence bond (VB) theory, together with the quantum theory of atoms in molecules (QTAIM) approach, have been used to probe the intramolecular hydrogen bonding interactions in a series of representative systems of resonance-assisted hydrogen bonds (RAHBs). RAHB is characteristic of the cooperativity between the π-electron delocalization and hydrogen bonding interactions and is identified in many biological systems. While the deactivation of the π resonance in these RAHB systems by the use of the BLW method is expected to considerably weaken the hydrogen bonding strength, little change on the topological properties of electron densities at hydrogen bond critical points (HBCPs) is observed. This raises a question of whether the QTAIM topological parameters can be an effective measure of hydrogen bond strength.     

Improvement to the hypervirial theorem and Kramer's rule for the calculation of central potential matrix elements

Ameliorated recurrence relations for the calculation of matrix elements of central potential wavefunctions are presented. These were obtained by using the hypervirial theorem with V(r) three-dimensional potentials and f (r) arbitrary functions. This procedure leads to a generalization of the usual l = l′ diagonal three-dimensional and l == 0 one-dimensional hypervirial relations of first and second class. The use of this kind of generalization to the calculation of r^k integrals, allows one to obtain all off-diagonal recursion formulas for any V(r). Besides, for hydrogenic wavefunctions one gets to equations that reduce to the usual Kramer's rule as a particular diagonal case. The proposed approach can be straightforwardly extended to obtain recurrence relations for the calculation of two center integrals.