QTAIM as a research programme: a reply to Shahbazian

In this paper I briefly reply to Shant Shahbazian’s comments on my paper “Austere quantum mechanics as a reductive basis for chemistry” and argue that quantum theory of atoms in molecules can be characterised as a research programme in the theories of chemistry. I also explore the areas in which Shahbazian and me agree and disagree.     

Atoms in molecules as non-overlapping, bounded, space-filling open quantum systems

The quantum theory of atoms in molecules (QTAIM) uses physics to define an atom and its contribution to observable properties in a given system. It does so using the electron density and its flow in a magnetic field, the current density. These are the two fields that Schrödinger said should be used to explain and understand the properties of matter. It is the purpose of this paper to show how QTAIM bridges the conceptual gulf that separates the observations of chemistry from the realm of physics and do so in a manner that is both rigorous and conceptually simple. Since QTAIM employs real measurable fields, it enables one to present the findings of complex quantum mechanical calculations in a pictorial manner that isolates the essential physics. The time has arrived for a sea change in our attempts to predict and classify the observations of chemistry, time to replace the use of simplified and arbitrary models with the full predictive power of physics, as applied to an atom in a molecule.     

Atoms and bonds in molecules and chemical explanations

The concepts of atoms and bonds in molecules which appeared in chemistry during the nineteenth century are unavoidable to explain the structure and the reactivity of the matter at a chemical level of understanding. Although they can be criticized from a strict reductionist point of view, because neither atoms nor bonds are observable in the sense of quantum mechanics, the topological and statistical interpretative approaches of quantum chemistry (quantum theory of atoms in molecules, electron localization function and maximum probability domain) provide consistent definitions which accommodate chemistry and quantum mechanics.     

The Atomic Partial Charges Arboretum: Trying to See the Forest for the Trees

Atomic partial charges are among the most commonly used interpretive tools in quantum chemistry. Dozens of different ‘population analyses’ are in use, which are best seen as proxies (indirect gauges) rather than measurements of a ‘general ionicity’. For the GMTKN55 benchmark of nearly 2,500 main-group molecules, which span a broad swathe of chemical space, some two dozen different charge distributions were evaluated at the PBE0 level near the 1-particle basis set limit. The correlation matrix between the different charge distributions exhibits a block structure; blocking is, broadly speaking, by charge distribution class. A principal component analysis on the entire dataset suggests that nearly all variation can be accounted for by just two ‘principal components of ionicity’: one has all the distributions going in sync, while the second corresponds mainly to Bader QTAIM vs. all others. A weaker third component corresponds to electrostatic charge models in opposition to the orbital-based ones. The single charge distributions that have the greatest statistical similarity to the first principal component are iterated Hirshfeld (Hirshfeld-I) and a minimal-basis projected modification of Bickelhaupt charges. If three individual variables, rather than three principal components, are to be identified that contain most of the information in the whole dataset, one representative for each of the three classes of Corminboeuf et al. is needed: one based on partitioning of the density (such as QTAIM), a second based on orbital partitioning (such as NPA), and a third based on the molecular electrostatic potential (such as HLY or CHELPG).     

A neglected aspect of the puzzle of chemical structure: how history helps

Intra-molecular connectivity (that is, chemical structure) does not emerge from computations based on fundamental quantum-mechanical principles. In order to compute molecular electronic energies (of C₃H₄ hydrocarbons, for instance) quantum chemists must insert intra-molecular connectivity “by hand.” Some take this as an indication that chemistry cannot be reduced to physics: others consider it as evidence that quantum chemistry needs new logical foundations. Such discussions are generally synchronic rather than diachronic—that is, they neglect ‘historical’ aspects. However, systems of interest to chemists generally are metastable. In many cases chemical systems of a given elemental composition may exist in any one of several different metastable states depending on the history of the system. Molecular structure generally depends on contingent historical circumstances of synthesis and separation, rather than solely or mainly on relative energies of alternative stable states, those energies in turn determined by relationships among components. Chemical structure is usually ‘kinetically-determined’ rather than ‘thermodynamically-determined.’ For instance, cyclical hydrocarbon ring-systems (as in cyclopropene) are produced only in special circumstances. Adequate theoretical treatments must take account of the persistent effects of such contingent historical events whenever they are relevant—as they generally are in chemistry.     

Toward universal substituent constants: Model chemistry sensitivity of descriptors from the quantum theory of atoms in molecules

The quantum theory of atoms in molecules (QTAIM) provides a theoretical foundation to determine the properties of functional groups through additive atomic contributions. Many studies have used QTAIM in their analyses with a variety of electronic structure methods, but it is unknown if the properties measured using one model chemistry, the combination of the electronic structure method and basis set, can be compared to those measured by another. Here, we evaluate the sensitivity of QTAIM functional group and bond critical point properties using six functionals and seven basis sets. High-level B2PLYPD3-BJ/aug-cc-pV5Z reference values are provided for 116 functional groups and the property sensitivity with respect to these values are evaluated based on absolute deviations and by assessing linear relationships. Functional group properties, including charges, dipoles, quadrupoles and volumes, were found to be mostly insensitive to choice of computational model chemistry. However, due to structural and topological inconsistencies, the 6-31G(d) basis set is not recommended for use. Bond critical point properties varied with choice of model chemistry, but models incorporating hybrid functionals and triple-ζ basis sets provided values suitable for use in regression studies.     

Current trends in the development of A. M. Butlerov’s theory of chemical structure

The chemical structure concept developed by A. M. Butlerov supplemented by the views on spatial (J. H. van’t-Hoff and J. A. Le Bel) and electronic (G. Lewis) configurations of molecules constitute the basis of the classical theory of chemical structure. The advent of quantum mechanics and development of the computer chemistry extended and enhanced the conceptual basis of theoretical chemistry, which nevertheless retains its independent value and cannot be reduced to direct physical definitions. The review deals with the evolution of the key concepts of the classical theory of chemical structure and introduction of new notions and approaches to analysis of the structure and reactivity problems, which is associated with the advent of the quantum mechanics and quantum chemistry views and methods.     

On the description of high-resolution experiments in molecular physics

We argue that all high-resolution experiments on small molecules some of which currently “belong” to chemistry and others to physics, should be brought together and discussed in a unified way using the best theory available, namely quantum mechanics, and that preconceptions about “molecular structure” should be avoided in this area of physical science.     

Matters are not so clear on the physical side

According to ontological reductionism, molecular chemistry refers, at last, to the quantum ontology; therefore, the ontological commitments of chemistry turn out to be finally grounded on quantum mechanics. The main problem of this position is that nobody really knows what quantum ontology is. The purpose of this work is to argue that the confidence in the existence of the physical entities described by quantum mechanics does not take into account the interpretative problems of the theory: in the discussions about the relationship between chemistry and physics, difficulties are seen only on the side of chemistry, whereas matters highly controversial on the side of physics are taken for granted. For instance, it is usually supposed that the infinite mass limit in the Born-Oppenheimer approximation leads by itself to the concept of molecular framework used in molecular chemistry. We will argue that this assumption is implicitly based on an interpretative postulate for quantum mechanics, which, in turn, runs into difficulties when applied to the explanation of the simplest model of the hydrogen atom.     

Molecular insight into the interaction mechanisms of amino‐2H‐imidazole derivatives with BACE1 protease: A QM/MM and QTAIM study

In this study, we described quantitatively the interactions between two new amino‐2H‐imidazole inhibitors ((R)‐1t and (S)‐1m) and BACE1 using a hybrid quantum mechanics‐molecular mechanical (QM/MM) method together with a quantum theory of atoms In Molecules (QTAIM) analysis. Our computational calculations revealed that the binding affinity of these compounds is mostly related to the amino‐2H‐imidazole core, which interact tightly with the aspartate dyad of the active site. The interactions were stronger when the inhibitors presented a bulky substituent with a hydrogen bond acceptor motif pointing toward Trp76, such as the 3,5‐dimethyl‐4‐methoxyphenyl group of compound (S)‐1m. Furthermore, the QTAIM analysis revealed that many hydrophobic interactions complement cooperatively the hydrogen bond which is not present when compound (R)‐1t is bound to the enzyme. The combined QM/MM‐QTAIM analysis allows identifying the interactions that account for the activity difference between compounds, even at a nanomolar range.     

Actinium coordination chemistry: A density functional theory study with monodentate and bidentate ligands

In the current study, the coordination chemistry of nine-coordinate Ac(III) complexes with 35 monodentate and bidentate ligands was investigated using density functional theory (DFT) in terms of their geometries, charges, reaction energies, and bonding interactions. The energy decomposition analysis with naturals orbitals for chemical valence (EDA-NOCV) and the quantum theory of atoms in molecules (QTAIM) were employed as analysis methods. Trivalent Ac exhibits the highest affinities toward hard acids (such as charged oxophilic donors, fluoride), so its classification as a hard acid is justified. Natural population analysis quantified the involvement of 5f orbitals on Ac to be about 30% of total valence electron natural configuration indicating that Ac is a member of the actinide series. Pearson correlation coefficients were used to study the pairwise correlations among the bond lengths, ΔG reaction energies, charges on Ac and donor atoms, and data from EDA-NOCV and QTAIM. Strong correlations and anticorrelations were found between Voronoi charges on donor atoms with ΔG, EDA-NOCV interaction energies and QTAIM bond critical point densities.     

Early impact of quantum physics on chemistry: George Hevesy’s work on rare earth elements and Michael Polanyi’s absorption theory

After Heitler and London published their pioneering work on the application of quantum mechanics to chemistry in 1927, it became an almost unquestioned dogma that chemistry would soon disappear as a discipline of its own rights. Reductionism felt victorious in the hope of analytically describing the chemical bond and the structure of molecules. The old quantum theory has already produced a widely applied model for the structure of atoms and the explanation of the periodic system. This paper will show two examples of the entry of quantum physics into more classical fields of chemistry: inorganic chemistry and physical chemistry. Due to their professional networking, George Hevesy and Michael Polanyi found their ways to Niels Bohr and Fritz London, respectively, to cooperate in solving together some problems of classical chemistry. Their works on rare earth elements and adsorption theory throws light to the application of quantum physics outside the reductionist areas. They support the heuristic and persuasive value of quantum thinking in the 1920–1930s. Looking at Polanyi’s later oeuvre, his experience with adsorption theory could be a starting point of his non-justificationist philosophy.     

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.     

How big are atoms in crystals?

Atoms and bonds are central concepts in structural chemistry, but neither are concepts that arise naturally from the physics of condensed phases. It is ironic that the internuclear distances in crystals that are readily measured depend on the sizes of atoms, but since atoms in crystals can be defined in many different ways, all of them arbitrary and often incompatible, there is no natural way to express atomic size. I propose a simple coherent picture of Atoms-in-Crystals which combines properties selected from three different physically sound definitions of atoms and bonds. The charge density of the free atom that is used to construct the procrystal is represented by a sphere of constant charge density having the quantum theory of atoms in molecules (QTAIM) bonded radius. The sum of these radii is equal to the bond length that correlates with the bond flux (bond valence) in the flux theory of the bond. The use of this model is illustrated by answering the question: How big are atoms in crystals? The QTAIM bonded radii are shown to be simple functions of two properties, the number of quantum shells in the atomic core and the flux of the bond that links neighbouring atoms. Various radii can be defined. The univalent bonded radius measures the intrinsic size of the atom and is the same for all cations in a given row of the periodic table, but the observed bonded radius depends also on the bond flux that reflects the chemical environment.     

The Quantum Mechanical Basis of Conceptual Chemistry

Summary. An experimentalist approaching theory for an understanding of conceptual chemistry that can be related to measurable properties, focuses on the electron density distribution. One finds in the topology of the electron density the definition of an atom, of the bonding between atoms, and of the boundary condition for the extension of quantum mechanics to an open system – to an atom in a molecule. This paper describes this approach, as it evolved from the failure of existing models to a study of molecular charge distributions and of how these studies resulted in the extension of quantum mechanics to an open system using the action principle.     

The Quantum Mechanical Basis of Conceptual Chemistry

An experimentalist approaching theory for an understanding of conceptual chemistry that can be related to measurable properties, focuses on the electron density distribution. One finds in the topology of the electron density the definition of an atom, of the bonding between atoms, and of the boundary condition for the extension of quantum mechanics to an open system – to an atom in a molecule. This paper describes this approach, as it evolved from the failure of existing models to a study of molecular charge distributions and of how these studies resulted in the extension of quantum mechanics to an open system using the action principle.     

Forced Bonding and QTAIM Deficiencies: A Case Study of the Nature of Interactions in He@Adamantane and the Origin of the High Metastability

Calculations within the framework of the interacting quantum atoms (IQA) approach have shown that the interactions of the helium atom with both tertiary, tC, and secondary, sC, carbon atoms in the metastable He@adamantane (He@adam) endohedral complex are bonding in nature, whereas the earlier study performed within the framework of Bader’s quantum theory of atoms in molecules (QTAIM) revealed that only He???tC interactions are bonding. The He???tC and He???sC bonding interactions are shown to be forced by the high pressure that the helium and carbon atoms exert upon each other in He@adam. The occurrence of a bonding interaction between the helium and sC atoms, which are not linked by a bond path, clearly shows that the lack of a bond path between two atoms does not necessarily indicate the lack of a bonding interaction, as is asserted by QTAIM. IQA calculations showed that not only the destabilization of the adamantane cage, but also a huge internal destabilization of the helium atom, contribute to the metastability of He@adam, these contributions being roughly equal. This result disproves previous opinions based on QTAIM analysis that only the destabilization of the adamantane cage accounts for the endothermicity of He@adam. Also, it was found that there is no homeomorphism of the ρ( r ) and ‐v( r ) fields of He@adam. Comparison of the IQA and QTAIM results on the interactions in He@adam exposes other deficiencies of the QTAIM approach. The reasons for the deficiencies in the QTAIM approach are analyzed.     

A DFT investigation on hydrogen- and halogen-bonding interactions in dichloroacetic acid: application of NMR-GIAO and Bader theories

A quantum chemistry investigation was carried out to examine hydrogen- and halogen-bonds properties in crystalline dichloroacetic acid (DCAA). We reported a systematic density functional theory study of the ¹⁷O, ³⁵Cl, and ¹H nuclear magnetic resonance (NMR) parameters in DCAA. Our results indicated that for those nuclei participated in the hydrogen- and halogen-bonding interactions; NMR parameters exhibit considerable changes on going from the isolated molecule model to the crystalline DCAA. Of course, the magnitude of these changes at each nucleus depends directly on its amount of contribution to the interactions. The topology of the electron density of O–H···O, C–H···O, Cl···Cl, and Cl···O interactions in solid DCAA was characterized using quantum theory of atoms in molecules (QTAIM). Based on QTAIM results, a partial covalent character is attributed to the O–H···O hydrogen bonds in DCAA, whereas all C–H···O, Cl···O, and Cl···Cl intermolecular contacts are weak and basically electrostatic in nature. Moreover, an approximate linear relationship seems to exist for each of the proton chemical shifts and anisotropies as a function of ρ_BCP.