Toward the multi-component quantum theory of atoms in molecules: a variational derivation

The general formalism of an extended quantum theory of atoms in molecules (QTAIM) dealing with the multi-component quantum systems, composed of various types of quantum particles, is disclosed in this contribution. This novel methodology, termed as the multi-component QTAIM (MC-QTAIM), is able to deal with non-adiabatic ab initio wavefunctions extracting atoms in molecules quantifying their properties. It can also be applied to elucidate the AIM structure of exotic species and bound quantum systems consisting of fundamental elementary particles like positrons and muons. The formalism is based on the previously disclosed density combination idea that is extended to derive the multi-component subsystem hypervirial theorem as well as the extended subsystem energy functional. Through the extended subsystem variational procedure, inspired from Schrödinger’s original variational principle, the surface terms containing the flux of the current property densities are derived. Accordingly, the extended Gamma field is introduced during this variational procedure that is used as the basic scalar field in the topological analysis yielding atoms in molecules and their real space boundaries. The Gamma field is central to the MC-QTAIM, replacing the usual one-electron density employed in the orthodox QTAIM and corresponding topological analysis. Through the multi-component hypervirial theorem, various regional theorems are derived which are then used to quantify the mechanical properties of atoms in molecules; these include the force, virial, torque, power, continuity and current theorems. In order to demonstrate the capability of the formalism, isotopically asymmetric hydrogen molecules, HD, HT and DT as well as YX systems (Y = ⁶Li, ⁷Li; X = H, D, T) composed of electrons and two different nuclei, all treated equally as quantum waves instead of clamped particles, are analyzed within context of the MC-QTAIM. The resulting computational analysis demonstrates that the MC-QTAIM is able to yield reasonable topological structures similar to those observed previously for diatomic species within context of the orthodox QTAIM. The asymmetrical nature of these species, inherent in their non-Born–Oppenhiemer wavefunctions, manifests itself clearly in the MC-QTAIM analysis yielding two distinguishable atomic basins with different properties. These differences are rationalized generally by the observed electron transfer from one basin to the other. Finally, some possible future theoretical extensions are considered briefly.     

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     

The quantum theory of atoms in positronic molecules: A case study on diatomic species

This article presents the first systematic study of a series of diatomic positronic species using the recently proposed regional approach: the quantum theory of atoms in positronic molecules (QTAIPM). This survey includes the LiH,e⁺, NaH,e⁺, LiF,e⁺, NaF,e⁺, BeO,e⁺, MgO,e⁺, CN^?,e⁺, and OH^?,e⁺ species as typical examples. The computational algorithm of the whole analysis is communicated and reviewed in detail. The topological analysis of the joint density distribution reveals topological structures similar to those observed for the purely electronic systems; that is, each system decomposes into two quantum atoms. By considering some of the regional properties of these quantum atoms, it is demonstrated that the positron affects them seriously through two different mechanisms: direct and indirect contributions, the latter refers to electronic and geometric relaxations. The computational results clearly reveal the fact that the regional properties of the quantum atoms of positronic molecules are not deducible from their purely electronic counterparts; thus, an independent analysis is required for each positronic molecule. The positronic population is considered as a typical regional property showing that the attachment of a positron to a purely electronic system enhances the polarization of the electronic distribution. The concept of regional positron affinities is also introduced and discussed as a nonroutine application of the QTAIPM. The results of this article set the stage for further study on the quantum atoms of polyatomic positronic species. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

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     

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.     

Performance of the density matrix functional theory in the quantum theory of atoms in molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.     

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.     

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.     

Chemical bonding in view of electron charge density and kinetic energy density descriptors

Stalke's dilemma, stating that different chemical interpretations are obtained when one and the same density is interpreted either by means of natural bond orbital (NBO) and subsequent natural resonance theory (NRT) application or by the quantum theory of atoms in molecules (QTAIM), is reinvestigated. It is shown that within the framework of QTAIM, the question as to whether for a given molecule two atoms are bonded or not is only meaningful in the context of a well‐defined reference geometry. The localized‐orbital‐locator (LOL) is applied to map out patterns in covalent bonding interaction, and produces results that are consistent for a variety of reference geometries. Furthermore, LOL interpretations are in accord with NBO/NRT, and assist in an interpretation in terms of covalent bonding. © 2008 Wiley Periodicals, Inc.J Comput Chem, 2009.     

Novel properties from experimental charge densities: an application to the zwitterionic neurotransmitter taurine

The charge distribution of taurine (2-aminoethane-sulfonic acid) is revisited by using an orbital-based method that describes the density in a fixed molecular orbital basis with variable orbital occupation numbers. A new neutron data set is also employed to explore whether this improves the deconvolution of thermal motion and charge density. A range of molecular properties that are novel for experimentally determined charge densities are computed, including Weinhold population analysis, Mayer bond orders, and local kinetic energy densities, in addition to charge topological analysis and quantum theory of atoms-in-molecules (QTAIM) integrated properties. The ease with which a distributed multipole analysis can be performed on the fitted density matrix makes it straightforward to compute molecular moments, the lattice energy, and the electrostatic interaction energies of molecules removed from the crystal. Results are compared with high-level (QCISD) gas-phase calculations and band structure calculations employing density functional theory. Finally, the avenues available for extending the range of molecular properties that can be calculated from experimental charge densities still further using this approach are discussed.     

On the connections between the quantum theory of atoms in molecules (QTAIM) and density functional theory (DFT): a letter from Richard F. W. Bader to Lou Massa

A 31-year-old letter from Professor Richard F. W. Bader to Professor Lou Massa outlining the connections between the quantum theory of atoms in molecules (QTAIM) and density functional theory (DFT) especially with regard to the first Hohenberg-Kohn theorem is brought to light. This connection has not often been the topic of such a focused review by Bader and is presented here for the first time. The scientific importance of this letter is, in the opinion of the presenter, as timely today as it was back then in 1986. In Bader’s own opening words: “... that if I sent you a summary of what I think are the important connections between our work and density functional theory, ...”. He then takes us in a grand tour of the foundations of QTAIM culminating into the antecedents of a paper he later published with Professor Pierre Becker, whereby the Hohenberg-Kohn theorem is shown to operate at the level of an atom-in-a-molecule. Bader closes his letter by suggesting to Massa: “Study these two charge distributions – they are proof of the theorem of Hohenberg and Kohn”. By that Bader meant that when the charge distributions of two atoms or groups are identical within a given precision, then the kinetic and total energy contributions of these atoms to the corresponding molecular quantities are also identical. It is revealing to follow the intellectual threads weaved by Bader which provides us with a glimpse of his thought processes and intuition that guided him to some of his key discoveries. The lucidity, rigor, and clarity characteristic of Bader and the informality of style of a letter makes it of pedagogic and historic interest.

     

Revisiting the foundations of the quantum theory of atoms in molecules: Some open problems

In a series of papers in the last 10 years, various aspects of the mathematical foundations of the quantum theory of atoms in molecules have been considered by this author and his coworkers in some details. Although these considerations answered part of the questions raised by some critics on the mathematical foundations of the quantum theory of atoms in molecules, however, new mathematical problems also emerged during these studies that were reviewed elsewhere [Sh. Shahbazian Int. J. Quantum Chem. 2011 , 111, 4497.]. Beyond mathematical subtleties of the formalism that were the original motivation for initial exchanges and disputes, the questions raised by critics had a constructive effect and prompted the author to propose a novel extension of the theory, now called the multi‐component quantum theory of atoms in molecules [M. Goli, Sh. Shahbazian Theor. Chem. Acc. 2013 , 132, 1365.]. Taking this background into account, in this paper a new set of open problems is put forward that the author believes proper answers to these questions, may open new doors for future theoretical developments of the quantum theory of atoms in molecules. Accordingly, rather than emphasizing on the rigorous mathematical formulation, the practical motivations behind proposing these questions are discussed in detail and the relevant literature are reviewed while when possible, evidence and routes to answers are also provided. The author hopes that proposing these open questions as a compact package may motivate more mathematically oriented people to participate in future developments of the quantum theory of atoms in molecules and its multi‐component version.     

Single determinant N‐representability and the kernel energy method applied to water clusters

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single‐determinant N‐representable one‐body density matrix. A kernel expansion for the one‐body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single‐determinant N‐representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N‐representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.     

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.     

Chemical Bonds and Spin State Splittings in Spin Crossover Complexes. A DFT and QTAIM Analysis

Summary. Density functional theory (DFT) calculations have been performed for the high-spin (HS) and low-spin (LS) isomers of a series of iron(II) spin crossover complexes with nitrogen ligands. The calculated charge densities have been analyzed in the framework of the quantum theory of atoms in molecules (QTAIM). For a number of iron(II) complexes with substituted tris(pyrazolyl) ligands the energy difference between HS and LS isomers, the spin state splitting, has been decomposed into atomic contributions in order to rationalize changes of the spin state splitting due to substituent effects.     

Biphenyl: A stress tensor and vector‐based perspective explored within the quantum theory of atoms in molecules

We use quantum theory of atoms in molecules (QTAIM) and the stress tensor topological approaches to explain the effects of the torsion φ of the C‐C bond linking the two phenyl rings of the biphenyl molecule on a bond‐by‐bond basis using both a scalar and vector‐based analysis. Using the total local energy density H( r _b), we show the favorable conditions for the formation of the controversial H–H bonding interactions for a planar biphenyl geometry. This bond‐by‐bond QTAIM analysis is found to be agreement with an earlier alternative QTAIM atom‐by‐atom approach that indicated that the H–H bonding interaction provided a locally stabilizing effect that is overwhelmed by the destabilizing role of the C‐C bond. This leads to a global destabilization of the planar biphenyl conformation compared with the twisted global minimum. In addition, the H( r _b) analysis showed that only the central torsional C‐C bond indicated a minimum for a torsion φ value coinciding with that of the conventional global energy minimum. The H–H bonding interactions are found to be topologically unstable for any torsion of the central C‐C bond away from the planar biphenyl geometry. Conversely, we demonstrate that for 0.0° < φ < 39.95° there is a resultant increase in the topological stability of the C nuclei comprising the central torsional C‐C bond. Evidence is found of the effect of the H–H bonding interactions on the torsion φ of the central C‐C bond of the biphenyl molecule in the form of the QTAIM response β of the total electronic charge density ρ( r _b). Using a vector‐based treatment of QTAIM we confirm the presence of the sharing of chemical character between adjacent bonds. In addition, we present a QTAIM interpretation of hyperconjugation and conjugation effects, the former was quantified as larger in agreement with molecular orbital (MO) theory. The stress tensor and the QTAIM H atomic basin path set areas are independently found to be new tools relevant for the incommensurate gas to solid phase transition occurring in biphenyl for a value of the torsion reaction coordinate φ ≈ 5°. © 2015 Wiley Periodicals, Inc.     

Conformational analysis of 18-azacrown-6 and its bonding with late first transition series divalent metals: insight from DFT combined with NPA and QTAIM analyses

Density functional theory calculations, together with quantum theory of atoms in molecules (QTAIM) analyses, have been performed to investigate 18-azacrown-6 complexes of the high-spin late first transition series divalent metal ions in the gas phase and, in some cases, in aqueous solution simulated by a polarizable continuum model. Six intramolecular H-H bonding interactions in the meso-complexes are found to arise from folding of the ligand upon its electrostatic interaction with the metal ions, which are largely absent in the lowest-energy C(2h) conformer of the free ligand. The ligand-to-metal charge transfer obtained from QTAIM analysis, among other things, is found to be an important factor that controls the stability of these complexes. The inter-relationship between the ligand preorganization energy, the zero-point corrected formation energy of the metal complexes, and the H-H bonding pair distances, as well as the dependence of the electron density and the total energy density at the H-H bond critical points on the H-H bonding pair distances, provides a physical basis for understanding and explaining the stabilizing nature of these closed-shell interactions, which are often viewed as steric clashes that lead to complex destabilization.