On the physical role of exchange in the formation of an intramolecular bond path between two electronegative atoms

Bader's Topological Bond Path Does Not Necessarily Indicate Stabilizing Interaction-Proof Studies Based on the Ng@[3n]cyclophane Endohedral Complexes

According to Bader's quantum theory of atoms in molecules (QTAIM), the simultaneous presence of a bond path and the corresponding bond critical point between any two atoms is both a necessary and sufficient condition for the atoms to be bonded to one another. In principle, this means that this pair of atoms should make a stabilizing contribution to the molecular system. However, the multitude of so-called counterintuitive bond paths strongly suggests that this statement is not necessarily true. Particularly 'troublesome' are endohedral complexes, in which encapsulation-enforced proximity between the trapped guest (e.g., an atom) and the host's cage system usually 'produces' many counterintuitive bond paths. In the author's opinion, the best evidence to demonstrate the repulsive nature of the intra-cage guest⋯host interaction is the use of some trapping systems containing small escape channels and then showing that the initially trapped entity spontaneously escapes outside the host's cage during geometry optimization of the initially built guest@host endohedral complex. For this purpose, a group of 24 Ng@[3n]cyclophane (3≤n≤6) endohedral complexes is used. As a result, arguments are presented showing that Bader's topological bond path does not necessarily indicate a stabilizing interaction.

Exact decompositions of the total KS-DFT exchange-correlation energy into one- and two-center terms

In the so-called Interacting Quantum Atoms (IQA) approach, the molecular energy is numerically decomposed as a sum of atomic and diatomic contributions. While proper formulations have been put forward for both Hartree-Fock and post-Hartree-Fock wavefunctions, this is not the case for the Kohn-Sham density functional theory (KS-DFT). In this work, we critically analyze the performance of two fully additive approaches for the IQA decomposition of the KS-DFT energy, namely, the one from Francisco et al., which uses atomic scaling factors, and that from Salvador and Mayer based upon the bond order density (SM-IQA). Atomic and diatomic exchange-correlation (xc) energy components are obtained for a molecular test set comprising different bond types and multiplicities and along the reaction coordinate of a Diels-Alder reaction. Both methodologies behave similarly for all systems considered. In general, the SM-IQA diatomic xc components are less negative than the Hartree-Fock ones, which is in good agreement with the known effect of electron correlation upon (most) covalent bonds. In addition, a new general scheme to minimize the numerical error of the sum of two-electron energy contributions (i.e., Coulomb and exact exchange) in the framework of overlapping atoms is described in detail.

The Ultrashort Spike-Ring Interaction in Substituted Iron Maiden Molecules

The in forms of molecular iron maidens are known for their unique ultrashort interaction between the apical hydrogen atom or its small substituent and the surface of the benzene ring. It is generally believed that this forced ultrashort X⋯π contact is associated with high steric hindrance, which is responsible for specific properties of iron maiden molecules. The main aim of this article is to investigate the influence of significant charge enrichment or depletion of the benzene ring on the characteristics of the ultrashort C-X⋯π contact in iron maiden molecules. For this purpose, three strongly electron-donating (-NH2) or strongly electron-withdrawing (-CN) groups were inserted into the benzene ring of in-[34,10][7]metacyclophane and its halogenated (X = F, Cl, Br) derivatives. It is shown that, despite such extremely electron-donating or electron-accepting properties, the considered iron maiden molecules surprisingly reveal quite high resistance to changes in electronic properties.

Real-Space Interpretation of Interatomic Charge Transfer and Electron Exchange Effects by Combining Static and Kinetic Potentials and Associated Vector Fields

Intricate behaviour of one-electron potentials from the Euler equation for electron density and corresponding gradient force fields in crystals was studied. Channels of locally enhanced kinetic potential and corresponding saddle Lagrange points were found between chemically bonded atoms. Superposition of electrostatic ϕ e s r and kinetic ϕ k r potentials and electron density ρ r allowed partitioning any molecules and crystals into atomic ρ - and potential-based ϕ -basins; ϕ k -basins explicitly account for the electron exchange effect, which is missed for ϕ e s -ones. Phenomena of interatomic charge transfer and related electron exchange were explained in terms of space gaps between zero-flux surfaces of ρ - and ϕ -basins. The gap between ϕ e s - and ρ -basins represents the charge transfer, while the gap between ϕ k - and ρ -basins is a real-space manifestation of sharing the transferred electrons caused by the static exchange and kinetic effects as a response against the electron transfer. The regularity describing relative positions of ρ -, ϕ e s -, and ϕ k - basin boundaries between interacting atoms was proposed. The position of ϕ k -boundary between ϕ e s - and ρ -ones within an electron occupier atom determines the extent of transferred electron sharing. The stronger an H⋅⋅⋅O hydrogen bond is, the deeper hydrogen atom's ϕ k -basin penetrates oxygen atom's ρ -basin, while for covalent bonds a ϕ k -boundary closely approaches a ϕ e s -one indicating almost complete sharing of the transferred electrons. In the case of ionic bonds, the same region corresponds to electron pairing within the ρ -basin of an electron occupier atom.

Non-covalent interactions from a Quantum Chemical Topology perspective

About half a century after its little-known beginnings, the quantum topological approach called QTAIM has grown into a widespread, but still not mainstream, methodology of interpretational quantum chemistry. Although often confused in textbooks with yet another population analysis, be it perhaps an elegant but somewhat esoteric one, QTAIM has been enriched with about a dozen other research areas sharing its main mathematical language, such as Interacting Quantum Atoms (IQA) or Electron Localisation Function (ELF), to form an overarching approach called Quantum Chemical Topology (QCT). Instead of reviewing the latter's role in understanding non-covalent interactions, we propose a number of ideas emerging from the full consequences of the space-filling nature of topological atoms, and discuss how they (will) impact on interatomic interactions, including non-covalent ones. The architecture of a force field called FFLUX, which is based on these ideas, is outlined. A new method called Relative Energy Gradient (REG) is put forward, which is able, by computation, to detect which fragments of a given molecular assembly govern the energetic behaviour of this whole assembly. This method can offer insight into the typical balance of competing atomic energies both in covalent and non-covalent case studies. A brief discussion on so-called bond critical points is given, highlighting concerns about their meaning, mainly in the arena of non-covalent interactions.

The physical nature of the ultrashort spike-ring interaction in iron maiden molecules

The so-called 'iron maiden' molecules belong to one of the most interesting subgroups of cyclophanes due to the presence of the ultrashort interaction between the CX apical bond and the benzene ring. This article presents an in-depth theoretical study of 16 'iron maiden' molecules, in which X = H, F, Cl or Br and the side chains are of various lengths and types: CSC, CSCC, CCC, and CCCC. It is shown that the H → F → Cl → Br substitution leads to a significant expansion of the 'iron maiden' molecule. Shorter chains lead to more pronounced effects, while insertion of sulfur atoms into the side chains lowers them. Structural changes are associated with an increase in energetic destabilization of X. Moreover, unlike for H, in the case of X = halogen, the out → in isomerization is energetically disadvantageous. The 'iron maiden' molecules are characterized by the presence of only three X⋯C_Ar bond paths. Particularly noteworthy are unusually large (even up to 32) values of the X⋯C_Ar bond ellipticity, which results from flat electron density distribution. The X⋯π interaction in each of the investigated 'iron maiden' molecule turned out to be multi-center, stabilizing and almost purely covalent in nature as indicated by the definitely dominant percentage (94.8%-101.6%) of the exchange-correlation energy. The spatial hindrance within the 'iron maiden' molecules appears to be not so much due to the X⋯π repulsion, but due to unfavorable steric interactions between X and the CC side bonds. It is also confirmed that some CH⋯HC interactions in aliphatic chains can be very weakly stabilizing.

Determining Repulsion in Cyclophane Cages

Superphane, i.e., [2.2.2.2.2.2](1,2,3,4,5,6)cyclophane, is a very convenient molecule in studying the nature of guest⋯host interactions in endohedral complexes. Nevertheless, the presence of as many as six ethylene bridges in the superphane molecule makes it practically impossible for the trapped entity to escape out of the superphane cage. Thus, in this article, I have implemented the idea of using the superphane derivatives with a reduced number of ethylene linkers, which leads to the [2n] cyclophanes where n<6. Seven such cyclophanes are then allowed to form endohedral complexes with noble gas (Ng) atoms (He, Ne, Ar, Kr). It is shown that in the vast majority of cases, the initially trapped Ng atom spontaneously escapes from the cyclophane cage, creating an exohedral complex. This is the best proof that the Ng⋯cyclophane interaction in endohedral complexes is indeed highly repulsive, i.e., destabilizing. Apart from the ‘sealed’ superphane molecule, endohedral complexes are only formed in the case of the smallest He atom. However, it has been shown that in these cases, the Ng⋯cyclophane interaction inside the cyclophane cage is nonbonding, i.e., repulsive. This highly energetically unfavorable effect causes the cyclophane molecule to ‘swell’.

Intrinsic Dynamic and Static Nature of π···π Interactions in Fused Benzene-Type Helicenes and Dimers, Elucidated with QTAIM Dual Functional Analysis

The intrinsic dynamic and static nature of the π···π interactions between the phenyl groups in proximity of helicenes 3-12 are elucidated with the quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA). The π···π interactions appear in C-∗-C, H-∗-H, and C-∗-H, with the asterisks indicating the existence of bond critical points (BCPs) on the interactions. The interactions of 3-12 are all predicted to have a p-CS/vdW nature (vdW nature of the pure closed-shell interaction), except for 2Cbay-∗-7Cbay of 10, which has a p-CS/t-HBnc nature (typical-HBs with no covalency). (See the text for definition of the numbers of C and the bay and cape areas). The natures of the interactions are similarly elucidated between the components of helicene dimers 6:6 and 7:7 with QTAIM-DFA, which have a p-CS/vdW nature. The characteristic electronic structures of helicenes are clarified through the natures predicted with QTAIM-DFA. Some bond paths (BPs) in helicenes appeared or disappeared, depending on the calculation methods. The static nature of Ccape-∗-Ccape is very similar to that of Cbay-∗-Cbay in 9-12, whereas the dynamic nature of Ccape-∗-Ccape appears to be very different from that of Cbay-∗-Cbay. The results will be a guide to design the helicene-containing materials of high functionality.

Does the Presence of a Bond Path Really Mean Interatomic Stabilization? The Case of the Ng@Superphane (Ng = He, Ne, Ar, and Kr) Endohedral Complexes

Using a fairly structurally flexible and, therefore, very suitable for this type of research, superphane molecule, we demonstrate that the inclusion of a noble gas atom (Ng = He, Ne, Ar, and Kr) inside it and, thus, the formation of the Ng@superphane endohedral complex, leads to its ‘swelling’. Positive values of both the binding and strain energies prove that encapsulation and in turn ‘swelling’ of the superphane molecule is energetically unfavorable and that the Ng⋯C interactions in the interior of the cage are destabilizing, i.e., repulsive. Additionally, negative Mayer Bond Orders indicate the antibonding nature of Ng⋯C contacts. This result in combination with the observed Ng⋯C bond paths shows that the presence of a bond path in the molecular graph does not necessarily prove interatomic stabilization. It is shown that the obtained conclusions do not depend on the computational methodology, i.e., the method and the basis set used. However, on the contrary, the number of bond paths may depend on the methodology. This is yet another disadvantageous finding that does not favor the treatment of bond paths on molecular graphs as indicators of chemical bonds. The Kr@superphane endohedral complex features one of the longest C–C bonds ever reported (1.753 Å).

On the Relationship between Hydrogen Bond Strength and the Formation Energy in Resonance-Assisted Hydrogen Bonds

Resonance-assisted hydrogen bonds (RAHB) are intramolecular contacts that are characterised by being particularly energetic. This fact is often attributed to the delocalisation of π electrons in the system. In the present article, we assess this thesis via the examination of the effect of electron-withdrawing and electron-donating groups, namely -F, -Cl, -Br, -CF3, -N(CH3)2, -OCH3, -NHCOCH3 on the strength of the RAHB in malondialdehyde by using the Quantum Theory of Atoms in Molecules (QTAIM) and the Interacting Quantum Atoms (IQA) analyses. We show that the influence of the investigated substituents on the strength of the investigated RAHBs depends largely on its position within the π skeleton. We also examine the relationship between the formation energy of the RAHB and the hydrogen bond interaction energy as defined by the IQA method of wave function analysis. We demonstrate that these substituents can have different effects on the formation and interaction energies, casting doubts regarding the use of different parameters as indicators of the RAHB formation energies. Finally, we also demonstrate how the energy density can offer an estimation of the IQA interaction energy, and therefore of the HB strength, at a reduced computational cost for these important interactions. We expected that the results reported herein will provide a valuable understanding in the assessment of the energetics of RAHB and other intramolecular interactions.

A Critical Overview of Current Theoretical Methods of Estimating the Energy of Intramolecular Interactions

This article is probably the first such comprehensive review of theoretical methods for estimating the energy of intramolecular hydrogen bonds or other interactions that are frequently the subject of scientific research. Rather than on a plethora of numerical data, the main focus is on discussing the theoretical rationale of each method. Additionally, attention is paid to the fact that it is very often possible to use several variants of a particular method. Both of the methods themselves and their variants often give wide ranges of the obtained estimates. Attention is drawn to the fact that the applicability of a particular method may be significantly limited by various factors that disturb the reliability of the estimation, such as considerable structural changes or new important interactions in the reference system.

Cooperativity in a cycloalkane-1,2/1,3-polyol corona: Topological hydrogen bonding in 1,2-diol motifs

A corona, consisting of 18 carbon atoms bearing 12 hydroxy groups in a continuous hydrogen-bonded chain, is built up by alternating degenerate conformations of alternating alkane-1,2-diol and 1,3-diol motifs. Geometries, proton nuclear magnetic resonance shifts and interaction energies for the dodecahydroxycyclo-octadecane and selected fragments are determined by density functional calculations at the B3LYP/6-311+G(d,p) level. Cooperative effects of O-H⋯O-H bonding are evident from the simple juxtaposition of these two motifs with a common OH group in butane-1,2,4-triol conformers. Bracketing a 1,2-diol motif with two 1,3-diol motifs in hexane-1,3,4,6-tetrol leads to a structure in which the 1,2-diol motif displays a bond critical point for hydrogen bonding. This is associated with enhancement of the shift of the hydrogen-bonded OH proton and of the corresponding H⋯O interaction energy. The full corona has a complete outer ring of O-H⋯O-H bond paths, and an inner ring of bond paths, due to C-H⋯H-C hydrogen-hydrogen bonding, which result in a central ring critical point. The topological O-H⋯O-H hydrogen bond, never seen in simple alkane-1,2-diols, is associated with cooperative enhancement of the H⋯O interaction energy, but this is not a necessary condition for a bond path: values for topological C-H⋯H-C hydrogen-hydrogen bonds can be as low as -0.4 kcal mol^-1 .

Interacting Quantum Atoms-A Review

The aim of this review is threefold. On the one hand, we intend it to serve as a gentle introduction to the Interacting Quantum Atoms (IQA) methodology for those unfamiliar with it. Second, we expect it to act as an up-to-date reference of recent developments related to IQA. Finally, we want it to highlight a non-exhaustive, yet representative set of showcase examples about how to use IQA to shed light in different chemical problems. To accomplish this, we start by providing a brief context to justify the development of IQA as a real space alternative to other existent energy partition schemes of the non-relativistic energy of molecules. We then introduce a self-contained algebraic derivation of the methodological IQA ecosystem as well as an overview of how these formulations vary with the level of theory employed to obtain the molecular wavefunction upon which the IQA procedure relies. Finally, we review the several applications of IQA as examined by different research groups worldwide to investigate a wide variety of chemical problems.

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.

Molecular Orbitals Support Energy-Stabilizing "Bonding" Nature of Bader's Bond Paths

Our MO-based findings proved a bonding nature of each density bridge (DB, or a bond path with an associated critical point, CP) on a Bader molecular graph. A DB pinpoints universal physical and net energy-lowering processes that might, but do not have to, lead to a chemical bond formation. Physical processes leading to electron density (ED) concentration in internuclear regions of three distinctively different homopolar H,H atom-pairs as well as classical C-C and C-H covalent bonds were found to be exactly the same. Notably, properties of individual MOs are internuclear-region specific as they (i) concentrate, deplete, or do not contribute to ED at a CP and (ii) delocalize electron-pairs through either in- (positive) or out-of-phase (negative) interference. Importantly, dominance of a net ED concentration and positive e^--pairs delocalization made by a number of σ-bonding MOs is a common feature at a CP. This feature was found for the covalently bonded atoms as well as homopolar H,H atom-pairs investigated. The latter refer to a DB-free H,H atom-pair of the bay in the twisted biphenyl (Bph) and DB-linked H,H atom-pairs (i) in cubic Li₄H₄, where each H atom is involved in three highly repulsive interactions (over +80 kcal/mol), and (ii) in a weak attractive interaction when sterically clashing in the planar Bph.

Cooperativity in alkane-1,2- and 1,3-polyols: NMR, QTAIM, and IQA study of O─H… OH and C─H… OH bonding interactions

Proton nuclear magnetic resonance chemical shifts and atom-atom interaction energies for alkanepolyols with 1,2-diol and 1,3-diol repeat units, and for their 1:1 pyridine complexes, are computed by density functional theory calculations. In the 1,3-polyols, based on a tG'Gg' repeat unit, the only important intramolecular hydrogen bonding interactions are O─H^… OH. By quantum theory of atoms in molecules analysis of the electron density, unstable bond and ring critical points are found for such interactions in 1,2-polyols with tG'g repeat units, from butane-1,2,3,4-tetrol onwards and in their pyridine complexes from propane-1,2,3-triol onwards. Several features (OH proton shifts and charges, and interaction energies computed by the interacting quantum atoms approach) are used to monitor the dependence of cooperativity on chain length: This is much less regular in 1,2-polyols than in 1,3-polyols and by most criteria has a higher damping factor. Well defined C─H^… OH interactions are found in butane-1,2,3,4-tetrol and higher members of the 1,2-polyol series, as well as in their pyridine complexes: There is no evidence for cooperativity with O─H^… OH bonding. For the 1,2-polyols, there is a tenuous empirical relationship between the existence of a bond critical point for O─H^… OH hydrogen bonding and the interaction energies of competing exchange channels, but the primary/secondary ratio is always less than unity.

Quantifying individual (anti)bonding molecular orbitals' contributions to chemical bonding

The shapes of molecular orbitals (MOs) in polyatomic molecules are often difficult for meaningful chemical interpretations. We report protocols to quantify contributions made by individual orbitals (molecular canonical and natural) of classical bonding, non-bonding or anti-bonding nature to (i) electron density into the inter-nuclear region and (ii) diatomic electron delocalization, DI(A,B). In other words, these protocols universally explain orbital's inputs to two fundamental and energy-lowering mechanisms of chemical bonding (interactions) and ease the chemical interpretation of their character in polyatomic molecules. They reveal that the MO and real-space density descriptions of the interactions are equivalent and, importantly, equally apply to all atom-pairs regardless if they are involved in a highly attractive or repulsive interaction. Hence, they not only remove ambiguity in chemical bonding interpretations (based on either MO or electron density approaches) but also demonstrate complementarity between the two such seemingly different techniques. Finally, our approach challenges some classical assumptions about MOs, such as the role of core electrons, the degree of bonding in antibonding MOs and the relative importance of frontier orbitals. Just as an example, we show that orthodox antibonding orbitals can make a significant contribution of a bonding nature to a classical covalent bond or major contribution to DI(A,B) of an intramolecular and highly repulsive HH interaction.

Nature of halogen-centered intermolecular interactions in crystal growth and design: Fluorine-centered interactions in dimers in crystalline hexafluoropropylene as a prototype

The wide occurrence of halogen-centered noncovalent interactions in crystal growth and design prompted this study, which includes a mini review of recent advances in the field. Particular emphasis is placed on providing compelling theoretical evidence of the formation of these interactions between sites of positive electrostatic potential, as well as between sites of negative electrostatic potential, localized on the electrostatic surfaces of the bound fluorine atoms in a prototypical system, hexafluoropropylene (C₃ F₆ ), upon its interaction with another same molecule to form (C₃ F₆ )₂ dimers. The existence of σ- and π-hole interactions is shown for the stable dimers. Even so, weakly bound interactions locally responsible in holding the molecular fragments together cannot and should not be overlooked since they are partly responsible for determining the overall geometry of the crystal. The results of combined quantum theory of atoms in molecules, molecular electrostatic surface potential, and reduced density gradient noncovalent interaction analyses showed that these latter interactions do indeed play a role in the stability and growth of crystalline C₃ F₆ itself and the (C₃ F₆ )₂ dimers. A symmetry adapted perturbation theory energy decomposition analysis leads to the conclusion that a great majority of the (C₃ F₆ )₂ dimers examined are the consequence of dispersion (and electrostatics), with nonnegligible contribution from polarization, which together competes with an exchange repulsion component to determine the equilibrium geometries. In a few structures of the (C₃ F₆ )₂ dimer, the fluorine is found to serve as a six-center five-bond donor/acceptor, as found for carbon in other systems (Malischewski and Seppelt, Angew. Chem. Int. Ed. 2017, 56, 368). © 2019 Wiley Periodicals, Inc.

On the Uselessness of Bond Paths Linking Distant Atoms and on the Violation of the Concept of Privileged Exchange Channels

We refer to frequently used determinants suggesting dominant interactions between distant atoms in various dimers. First of all, we show, against the still-prevailling opinion, that, in general, bond paths have nothing in common with dominant intermolecular interactions and therefore they are useless in such cases. Quite the contrary, reliable information about dominant intermolecular interactions can be obtained by means of electrostatic potential maps, which very convincingly explain mutual orientation of molecules in a dimer. For the first time, numerous examples of interactions that violate both the concept of privileged exchange channels proposed by Pendás and his collaborators as well as inequalities obtained by Tognetti and Joubert for the β parameter related to secondary interactions are presented. The possible cause of this violation is suggested. We also show that the so-called counterintuitive bond paths result from quite natural behavior of the electron density gradient vector, i. e. searching for those areas of space that are characterized by large values of electron density or the most expanded its distributions.

Does the Intra-Atomic Deformation Energy of Interacting Quantum Atoms Represent Steric Energy?

We show that the mutual, through-space compression of atomic volume experienced by approaching topological atoms causes an exponential increase in the intra-atomic energy of those atoms, regardless of approach orientation. This insight was obtained using the modern energy partitioning method called interacting quantum atoms (IQA). This behaviour is consistent for all atoms except hydrogen, which can behave differently depending on its environment. Whilst all atoms experience charge transfer when they interact, the intra-atomic energy of the hydrogen atom is more vulnerable to these changes than larger atoms. The difference in behaviour is found to be due to hydrogen's lack of a core of electrons, which, in heavier atoms, consistently provide repulsion when compressed. As such, hydrogen atoms do not always provide steric hindrance. In accounting for hydrogen's unusual behaviour and demonstrating the exponential character of the intra-atomic energy in all other atoms, we provide evidence for IQA's intra-atomic energy as a quantitative description of steric energy.

“Lp⋯synthon” interaction as a reason for the strong amplification of synthon-forming hydrogen bonds

A new type of interaction is presented.

A relative energy gradient (REG) study of the planar and perpendicular torsional energy barriers in biphenyl

Biphenyl is a prototype molecule, the study of which is important for a proper understanding of stereo-electronic effects. In the gas phase it has an equilibrium central torsion angle of ~ 45° and shows both a planar (0°) and a perpendicular (90°) torsional energy barrier. The latter is analysed for the first time. We use the newly proposed REG method, which is an exhaustive procedure that automatically ranks atomic energy contributions according to their importance in explaining the energy profile of a total system. Here, the REG method operates on energy contributions computed by the interacting quantum atoms method. This method is minimal in architecture and provides a crisp picture of well-defined and well-separated electrostatic, steric and exchange (covalent) energies at atomistic level. It is shown that the bond critical point occurring between the ortho-hydrogens in the planar geometry has been wrongly interpreted as a sign of repulsive interaction. A convenient metaphor of analysing football matches is introduced to clarify the role of a REG analysis.

On Electrostatics, Covalency, and Chemical Dashes: Physical Interactions versus Chemical Bonds

The increasing availability of real-space interaction energies between quantum atoms or fragments that provide a chemically intuitive decomposition of intrinsic bond energies into electrostatic and covalent terms [see, for instance, Chem. Eur. J. 2018, 24, 9101] provides evidence for differences between the physicist's concept of interaction and the chemist's concept of a bond. Herein, it is argued that, for the former, all types of interactions are treated equally, whereas, for the latter, only the covalent short-range interactions have actually been used to build intuition about chemical graphs and chemical bonds. This has led to the bonding role of long-range Coulombic terms in molecular chemistry being overlooked. Simultaneously, blind consideration of electrostatic terms in chemical bonding parlance may lead to confusion. The relationship between these concepts is examined herein, and some notes of caution on how to merge them are proposed.

FALDI-based criterion for and the origin of an electron density bridge with an associated (3,-1) critical point on Bader's molecular graph

The total electron density (ED) along the λ₂ -eigenvector is decomposed into contributions which either facilitate or hinder the presence of an electron density bridge (DB, often called an atomic interaction line or a bond path). Our FALDI-based approach explains a DB presence as a result of a dominating rate of change of facilitating factors relative to the rate of change of hindering factors; a novel and universal criterion for a DB presence is, thus, proposed. Importantly, facilitating factors show, in absolute terms, a concentration of ED in the internuclear region as commonly observed for most chemical bonds, whereas hindering factors show a depletion of ED in the internuclear region. We test our approach on four intramolecular interactions, namely (i) an attractive classical H-bond, (ii) a repulsive O⋅⋅⋅O interaction, (iii) an attractive Cl⋅⋅⋅Cl interaction, and (iv) an attractive CH⋅⋅⋅HC interaction. (Dis)appearance of a DB is (i) shown to be due to a "small" change in molecular environment and (ii) qualitatively and quantitatively linked with specific atoms and atom-pairs. The protocol described is equally applicable (a) to any internuclear region, (b) regardless of what kind of interaction (attractive/repulsive) atoms are involved in, (c) at any level of theory used to compute the molecular structure and corresponding wavefunction, and (d) equilibrium or nonequilibrium structures. Finally, we argue for a paradigm shift in the description of chemical interactions, from the ED perspective, in favor of a multicenter rather than diatomic approach in interpreting ED distributions in internuclear regions. © 2018 Wiley Periodicals, Inc.

Bond paths between distant atoms do not necessarily indicate dominant interactions

The goal of the article is to revive discussion on the interpretation of bond paths linking distant atoms, particularly tracing weak interactions in dimers. According to the Pendás' concept of privileged exchange channel, a bond path is formed between this pair of competing atoms, which is associated with larger value of the exchange energy. We point out that, due to the short-range nature of the exchange energy, bond paths linking distant atoms clearly become doubtful indicators of dominant intermolecular interactions, particularly if some other characteristics (geometric, spectroscopic, based on electrostatic parameters, etc.) indicate other intermolecular interactions as dominant. Several such cases are thoroughly investigated. We show that electrostatic parameters are much more reliable indicators of dominant intermolecular interactions than bond paths. Then, we pay attention that the presence of ("unexpected", i.e., not necessarily indicating dominant intermolecular interactions) bond paths between pairs of atoms featuring highly expanded charge distributions can be easily explained by visual exploration of isodensity contour plots. As always pointing in the direction of the steepest increase, the gradient vector of the electron density favors areas of its high values gaining higher exchange energy, yet being blind to highly electron deficient areas nearby, which, however, can quite often be involved in dominant intermolecular interactions as strongly suggested by many other bonding analysis. We also suggest that an interatomic component of Hellmann-Feynman force would most likely be the most reliable indicator of attractive or repulsive character of individual interatomic interaction. © 2018 Wiley Periodicals, Inc.

Using the Relative Energy Gradient Method with Interacting Quantum Atoms to Determine the Reaction Mechanism and Catalytic Effects in the Peptide Hydrolysis in HIV-1 Protease

The reaction mechanism in an active site is of the utmost importance when trying to understand the role that an enzyme plays in biological processes. In a recently published paper [Theor. Chem. Acc. 2017, 136, 86], we formalised the Relative Energy Gradient (REG) method for automating an Interacting Quantum Atoms (IQA) analysis. Here, the REG method is utilised to determine the mechanism of peptide hydrolysis in the aspartic active site of the enzyme HIV-1 Protease. Using the REG method along with the IQA approach we determine the mechanism of peptide hydrolysis without employing any arbitrary parameters and with remarkable ease (albeit at large computational cost: the system contains 133 atoms, which means that there are 17 689 individual IQA terms to be calculated). When REG and IQA work together it is possible to determine a reaction mechanism at atomistic resolution from data directly derived from quantum calculations, without arbitrary parameters. Moreover, the mechanism determined by this novel method gives concrete insight into how the active site residues catalyse peptide hydrolysis.

Fluorine Gauche Effect Explained by Electrostatic Polarization Instead of Hyperconjugation: An Interacting Quantum Atoms (IQA) and Relative Energy Gradient (REG) Study

We present an interacting quantum atoms (IQA) study of the gauche effect by comparing 1,2-difluoroethane, 1,2-dichloroethane, and three conformers of 1,2,3,4,5,6-hexafluorocyclohexane. In the 1,2-difluoroethane, the gauche effect is observed in that the gauche conformation is more stable than the anti, whereas in 1,2-dichloroethane the opposite is true. The analysis performed here is exhaustive and unbiased thanks to using the recently introduced relative energy gradient (REG) method [ Thacker , J. C. R. ; Popelier , P. L. A. Theor. Chem. Acc . 2017 , 136 , 86 ], as implemented in the in-house program ANANKE. We challenge the common explanation that hyperconjugation is responsible for the gauche stability in 1,2-difluoroethane and instead present electrostatics as the cause of gauche stability. Our explanation of the gauche effect is also is seen in other molecules displaying local gauche conformations, such as the recently synthesized "all-cis" hexafluorocyclohexane and its conformers where all the fluorine atoms are in the equatorial positions. Using our extension of the traditional IQA methodology that allows for the partitioning of electrostatic terms into polarization and charge transfer, we propose that the cause of gauche stability is 1,3 C···F electrostatic polarization interactions. In other words, if a number of fluorine atoms are aligned, then the stability due to polarization of nearby carbon atoms is increased.

Extension of the interacting quantum atoms (IQA) approach to B3LYP level density functional theory (DFT)

An interaction between two atoms, bonded or non-bonded, consists of interatomic contributions: electrostatic energy, exchange energy and electronic correlation energy. Together with the intra-atomic energy of an atom, these contributions are the basic components of the Interacting Quantum Atom (IQA) energy decomposition scheme. Here, we investigate IQA's proper use in conjunction with an explicit implementation of the B3LYP functional. The recovery of the total molecular energy from the IQA components is emphasised, for the first time. A systematic study of three model systems of biological relevance, N-methylacetamide (NMA), the doubly capped tripeptide GlyGlyGly and an alloxan dimer, shows the stabilization effect of B3LYP on most of the interatomic exchange energies (V) compared to their Hartree-Fock values. Diagrams of exchange energies versus interatomic distance show the clustering of interactions, one cluster for each 1,n (n = 1 to 6 where the atoms are separated by n - 1 bonds). The positioning of some V values outside their expected cluster marks interesting interactions.

Energetic Analysis of Conjugated Hydrocarbons Using the Interacting Quantum Atoms Method

A number of aromatic, antiaromatic, and nonaromatic organic molecules was analyzed in terms of the contributions to the electronic energy defined in the quantum theory of atoms in molecules and the interacting quantum atoms method. Regularities were found in the exchange and electrostatic interatomic energies showing trends that are closely related to those of the delocalization indices defined in the theory. In particular, the CC interaction energies between bonded atoms allow to rationalize the energetic stabilization associated with the bond length alternation in conjugated polyenes. This approach also provides support to Clar's sextet rules devised for aromatic systems. In addition, the H⋯H bonding found in some of the aromatic molecules studied was of an attractive nature, according to the stabilizing exchange interaction between the bonded H atoms. © 2017 Wiley Periodicals, Inc.

On the importance of intramolecular hydrogen bond cooperativity in d-glucose - an NMR and QTAIM approach

The idea that hydrogen bond cooperativity is responsible for the structure and reactivity of carbohydrates is examined. Density functional theory and gauge-including atomic orbital calculations on the known conformers of the α and β anomers of d-glucopyranose in the gas phase are used to compute proton NMR chemical shifts and interatomic distances, which are taken as criteria for probing intramolecular interactions. Atom-atom interaction energies are calculated by the interacting quantum atoms approach in the framework of the quantum theory of atoms in molecules. Association of OH1 in the counterclockwise conformers with a strong acceptor, pyridine, is accompanied by cooperative participation from OH2, but there is no significant change in the bonding of the two following 1,2-diol motifs. The OH6^... O5 (G-g+/cc/t and G+g-/cc/t conformers) or OH6^... O4 (Tg+/cc/t conformer) distance is reduced, and the OH6 proton is slightly deshielded. In the latter case, this shortening and the associated increase in the OH6-O4 interaction energy may be interpreted as a small cooperative effect, but intermolecular interaction energies are practically the same for all three conformers. In most of the pyridine complexes, one ortho proton interacts with the endocyclic oxygen O5. Analogous results are obtained when the clockwise conformer, G-g+/cl/g-, detected for the α anomer, and a hypothetical conformer, Tt/cl/g-, are complexed with pyridine through OH6. Generally, the cooperative effect does not go beyond the first two OH groups of a chain. Copyright © 2017 John Wiley & Sons, Ltd.

The ANANKE relative energy gradient (REG) method to automate IQA analysis over configurational change

Much chemical insight ultimately comes down to finding out which fragment of a total system behaves like the total system, in terms of an energy profile. A simple example is that of the water dimer, where this system is regarded as held together by a hydrogen bond. The hydrogen bond consists of two atoms (H···O), which energetically behave similarly to the total system (H₂O)₂. However, from a quantum mechanical point of view, each atom in the total system interacts with any other atom. Thus, the view that the hydrogen bond by itself governs the energetic stability of the water dimer needs rigorous justification. In this work, we propose a method that provides such a justification, in general, but only illustrated on the water dimer here. This method is based on the topological energy partitioning method called interacting quantum atoms (IQA). The method is implemented in the program ANANKE, which calculates correlations between the energy profile of the total system and those of subsystems (or fragments). ANANKE acts on the IQA energy contributions obtained for a sequence of full-system geometries controlled by a coordinate of interest (e.g. the O···H distance in the water dimer). Although applied only for the water dimer in this work, the method is general and able to explain the gauche effect, the torsional barrier in biphenyl, the arrow-pushing scheme of an enzymatic reaction (peptide hydrolysis in the HIV-1 Protease active site), and halogen-alkane nucleophilic substitution (S_N2) reactions. Those applications will appear elsewhere as separate and elaborated case studies; here we focus on the details of the ANANKE method and its justification, using the water dimer as a concrete case.

Nature ofE2X2σ(4c–6e) of theX---E—E---Xtype at naphthalene 1,8-positions and model, elucidated by X-ray crystallographic analysis and QC calculations with the QTAIM approach

The nature ofE2X2σ(4c–6e) of theX-*-E-*-E-*-Xtype is elucidated for 1-(8-XC10H6)E–E(C10H6X-8′)-1′ [(1)E,X= S, Cl; (2) S, Br; (3) Se, Cl; (4) Se, Br] after structural determination of (1), (3) and (4), together with modelA[MeX---E(H)—E(H)---XMe (E= S and Se;X= Cl and Br)]. The quantum theory of atoms-in-molecules dual functional analysis (QTAIM-DFA) is applied. The total electron energy densitiesHb(rc) are plottedversus Hb(rc) –Vb(rc)/2 for the interactions at the bond critical points (BCPs; *), whereVb(rc) show the potential energy densities at the BCPs. Data for the perturbed structures around the fully optimized structures are employed for the plots, in addition to those of the fully optimized structures. The plots were analysed using the polar coordinate (R, θ) representation of the data of the fully optimized structures. Data containing the perturbed structures were analysed by (θp, κp), where θpcorresponds to the tangent line of the plot and κpis the curvature. Whereas (R, θ) shows the static nature, (θp, κp) represents the dynamic nature of interactions.E-*-Eare all classified as shared shell (S) interactions for (1)–(4) and as weak covalent (Cov-w) in nature (S/Cov-w). The nature ofpureCS (closed shell)/typical-HB (hydrogen bond) with no covalency is predicted forE-*-Xin (1) and (3),regularCS/typical-HB nature with covalency is predicted for (4), and an intermediate nature is predicted for (2). The NBO energies evaluated forE-*-Xin (1)–(4) are substantially larger than those in modelAdue the shortened length at the naphthalene 1,8-positions. The nature ofE2X2of σ(4c–6e) is well elucidatedviaQTAIM-DFA.

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, Veexc , 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 Veexc and it emerged that for the hydrogen-carbon bond, the Veexc was distinct for each isotopic composition and varied in line with chemical expectations. Indeed, the introduction of Veexc paves the way for the investigation of vast numbers of structural and kinetic isotope effects within the context of the MC-QTAIM.

Torquoselectivity in Cyclobutene Ring Openings and the Interatomic Interactions That Control Them

Torquoselectivity has explained diasteromeric preferences of a number electrocyclic ring openings. The quantum theory of atoms in molecules (QTAIM), the electron localizability indicator (ELI-D), and the interacting quantum atoms (IQA) energy partition method are used to evaluate qualitatively and quantitatively the atomic interactions behind the torquoselectivity of a series of 3-substituted cyclobutenes. ELI-D topology and IQA energies show that the interaction between the distal terminus carbon atom of cyclobutene (C4) with the substituent at C3 (R5) in the transition state governs torquoselectivities. In the case of 3-borylcyclobutene, this interaction is so strong that a protocovalent bond is actually formed between B5 and C4. The evaluation of the interatomic energies allowed us to identify an additional interaction that contribute to a minor extent to the stabilization of the TS. Despite the fact that C4,R5 interaction is the main cause of the torquoselectivity, a bonding path (BP) between these two atoms was not observed. However, the lack of a BP between C4 and R5 does not mean that the topology of the electron density was not affected by the interaction of these two atoms. Surprisingly, we found a strong correlation between the density at the bond critical point (BCP) and the BP shape of C3-C4 breaking bond with the observed activation energies and torquoselectivities.

Association of symmetrical alkane diols with pyridine: DFT/GIAO calculation of 1 H NMR chemical shifts

Proton nuclear magnetic resonance (NMR) shifts of the free diol and of its 1 : 1 and 1 : 2 hydrogen-bonded complexes with pyridine have been computed for five symmetrical alkane diols on the basis of density functional theory, by applying the gauge-including atomic orbital method to geometry-optimized conformers. For certain conformers, intramolecular OH···OH interactions, evidenced by high NMR OH proton shifts, are further enhanced on going from the free diol to the corresponding 1 : 1 diol/pyridine complex. This is confirmed by atoms-in-molecules and non-covalent interaction plots. The computed OH and CH proton shifts for the diol and the two complexes correlate well with values obtained by analysing data from the NMR titration of the diols in benzene against pyridine. Shift values for the diols in neat pyridine are calculated by weighting the shifts of the various protons in the three forms (free diol, 1 : 1 and 1 : 2 diol/pyridine complexes) according to the experimentally determined association constants. The results are in good agreement with those observed, and after empirical scaling, the root mean square difference is 0.18 ppm. Copyright © 2016 John Wiley & Sons, Ltd.

Interacting quantum fragments-rooted preorganized-interacting fragments attributed relative molecular stability of the Be(II) complexes of nitrilotriacetic acid and nitrilotri-3-propionic acid

A method designed to investigate, on a fundamental level, the origin of relative stability of molecular systems using Be(II) complexes with nitrilotriacetic acid (NTA) and nitrilotri-3-propionic acid (NTPA) is described. It makes use of the primary and molecular fragment energy terms as defined in the IQA/F (Interacting Quantum Atoms/Fragments) framework. An extensive classical-type investigation, focused on single descriptors (bond length, density at critical point, the size of metal ion or coordination ring, interaction energy between Be(II) and a donor atom, etc.) showed that it is not possible to explain the experimental trend. The proposed methodology is fundamentally different in that it accounts for the total energy contributions coming from all atoms of selected molecular fragments, and monitors changes in defined energy terms (e.g., fragment deformation, inter- and intra-fragment interaction) on complex formation. By decomposing combined energy terms we identified the origin of relative stability of Be(II) (NTA) and Be(II) (NTPA) complexes. We found that the sum of coordination bonds' strength, as measured by interaction energies between Be(II) ion and donor atoms, favours Be(II) (NTA) but the binding energy of Be(II) ion to the entire ligand correlates well with experimental trend. Surprisingly, the origin of Be(II) (NTPA) being more stable is due to less severe repulsive interactions with the backbone of NTPA (C and H-atoms). This general purpose protocol can be employed not only to investigate the origin of relative stability of any molecular system (e.g., metal complexes) but, in principle, can be used as a predictive tool for, e.g., explaining reaction mechanism. © 2016 Wiley Periodicals, Inc.