Structure, energy, vibrational spectrum, and Bader's analysis of π⋯H hydrogen bonds and H^−δ⋯H^+δ dihydrogen bonds

Abstract

In this paper, the intermolecular structural study asserted by the vibrational analysis in the stretch frequencies of hydrogen bonds (π⋯H) and dihydrogen bonds (H^−δ⋯H^+δ) have definitively been revisited by means of calculations carried out by Density Functional Theory (DFT) and topological parameters derived from the classic treatise of the Quantum Theory of Atoms in Molecules (QTAIM). As a matter of fact the π⋯H hydrogen bond is formed between the hydrofluoric acid and the C≡C bond of the acetylene, but the QTAIM calculations revealed a distortion in this interaction due to the formation of the ternary complex C₂H₂⋯2(HF). Although the π bonds of ethylene (C₂H₄), propylene (C₂H₃(CH₃)), and t-butylene (C₂H₂(CH₃)₂) are considered proton acceptors, two hydrogen-bond types—π⋯H and C⋯H—can be observed. Over and above the analysis of the π hydrogen bonds, theoretical arguments also were used to discuss the red-shifts in the stretch frequencies of the binary dihydrogen complexes formed by BeH₂⋯HX with X = F, Cl, CN, and CCH. Although a vibrational blue-shift in the stretch frequency of the H–C bond of HCF₃ due to the formation of the BeH₂⋯HCF₃ dihydrogen complex was obtained, unmistakable red-shifts were detected in LiH⋯HCF₃, MgH₂⋯HCF₃, and NaH⋯HCF₃. Moreover, the alkali–halogen bonds were identified in relation to the formation of the trimolecular systems NaH⋯2(HCF₃) and NaH⋯2(HCCl₃). At last, theoretical calculations and QTAIM molecular integrations were used to study a novel class of dihydrogen-bonded complexes (mC₂H₅⁺ ⋯nMgH₂ with m = 1 or 2 and n = 1 or 2) based in the insight that MgH₂ can bind with the non-localized hydrogen H^+δ of the ethyl cation (C₂H₅⁺). In an overview, QTAIM calculations were applied to evaluate the molecular topography, charge density, as well as to interpret the shifted frequencies either to red or blue caused by the formation of the hydrogen bonds and dihydrogen bonds.

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Boaz Galdino de Oliveira

Boaz Galdino de Oliveira studied industrial chemistry at the State University of Paraíba (1999), obtained his D.Sc in chemical physics at the Federal University of Paraíba (2005) under the guidance of Professor Regiane C.M.U. Araújo, and spent four years in postdoctoral researches at the Federal University of Pernambuco (2009). Today he is adjunct professor at the Federal University of Bahia. He has experience in theoretical chemistry, acting on the following topics: hydrogen bonds, halogen bonds, dihydrogen bonds, stacking interactions, quantum interactions, quantum theory of atoms in molecules, chemometric investigations of density functional theory and ab initio methods.

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Introduction

It is widely known that the atomic properties determine a specific condition by which the bonded contacts or non-bonded interactions exist or coexist in a molecular system.^1–5 As such, it should be mentioned that the hydrogen bond is not only a feasible model of non-bonded interaction,^6–13 but its properties play a crucial role in studies of chemistry, biology and physics,^14–27 and thereby ascribe it as one of the most important interactions of nature. Moreover, the origin of the hydrogen bond is a singular topic and its application is mightily useful, such as for example in engineering,²⁸ medicinal chemistry with drug design,^29–31 polymers,^32–35 macromolecular research,^36–38 understanding of biochemical processes^39–42 such as enzymatic and organometallic catalysis,^43,44 synthesis,^45–47 molecular recognition,⁴⁸ development of raw materials to be used in industrial scale,⁴⁹ and molecular modelling of biological systems aided by spectroscopy parameters.^50,51 Nevertheless, some attention also has been dedicated to the applicability of hydrogen bonds in molecular modelling of cooperative systems,^52–63 wherein the origin of life based on DNA structure certainly is the most remarkable, incisive, and thorough investigation focus.^64–69 Notoriously all these researches provide a summary of the ‘state of the art for hydrogen bonds’,⁷⁰ although it must be remembered that its evidence is the greatest goal in any investigation type.^71–74 Over all these years, a lot of definitions for hydrogen bond^75–83 have been documented by famous scientists, such as Lewis,⁸⁴ Pauling,⁸⁵ Huggins,⁸⁶ Pimentel and McClellan,⁸⁷ Latimer and Rodebush⁸⁸ as well as Coulson and Danielson.⁸⁹ In the past, many other researchers designed empirical studies in order to understand the phenomena inherent to the hydrogen bond.^90–94 Although the experimental assays have helped decisively to characterize the formation of hydrogen bonds,^95–97 its first experimental evidence was documented only with the discovering of the X-ray technique at the beginning of the 20th century.^98,99

In order to capture the chemical genesis of the intermolecular interactions, the energy decomposition projected by Kitaura, Umeyama, Morokuma and co-authors,^100–104 Divide and Conquer (DC) elaborated by Merz Jr. et al.,^105–109 or even the Symmetry Adapted Perturbation Theory (SAPT) developed by Szalewicz and others,^110–117 are routinely applied in theoretical investigations aimed at examining the hydrogen bond through the quantification of the Coulomb energy, exchange term, dispersion forces, and charge transfer,^118,119 albeit it should be cited that the valence bond theory or quantum-chemical methods are traditional and important tools to investigate the non-bonded phenomenology.^120–126 In agreement with these assertions, there are three excellent study guides, namely are a book edited by Grabowski,¹²⁷ a review produced by Kojić-Prodić and Molćanov,¹²⁸ and a paper elaborated by Gilli and Gilli.¹²⁹ These works provide a detailed overview of hydrogen bonds, mainly the relationship between a proton donor and a region containing high electron density, and in addition the ideal condition to interact one with another is also stressed.^130–134 Such a condition has been crucial to define the hydrogen bonds, although, according to a IUPAC project entitled ‘categorizing hydrogen bonding and other intermolecular interactions’,^135,136 the hydrogen bonds are generated by two electronegative atoms (X and Y) interacting by means of a hydrogen atom. In other words, the interaction model is symbolized by X–H⋯Y with X and Y having been established to be N, O, and F,^137–139 in spite of some other elements that are also able to interact with proton donors.^140,141 Undoubtedly, that the hydrogen bond represents the most traditional noncovalent interaction profile ever seen,^142–148 but the property of proton donors (H^+δ) for binding with charge density sites,^149–151 such as lone electron pairs,^152–157 π or pseudo-π bonds,^158–161 surely is the guideline attributed to the elucidation of intermolecular interactions.^162–175

Some time ago, the Crabtree research group^176,177 discovered a remarkable intermolecular interaction type. One peculiar feature observed in this contact is the stable interaction between hydrides (H^−δ) and protons (H^+δ),^178,179 which intriguingly bind themselves and form the dihydrogen bond H^−δ⋯H^+δ.^180–185 This discovery led to the appearing of a new era in intermolecular systems researches with great prominence for the dihydrogen-bonded complexes, e.g., BH₃NH₃ dimer.¹⁸⁶ In this insight, it has been reported that BeH₂ operates well as proton receptor¹⁸⁷ by which the formation of complexes with rare gases¹⁸⁸ became reliable, or even in other situations such as reactions with transition metals,^189,190 for instance. In a research paper, Alkorta et al.¹⁹¹ have declared some tenuous differences between hydrogen bonds, dihydrogen bonds, and hydride bonds.^192–203 Namely, dihydrogen bonds are considered apart or even a special case of hydrogen bonds, whereas hydride bonds are formerly an opposite hydrogen bond. In order to ensure some theoretical insights about these interaction types, it seems to be fundamental that two basic features should be compulsorily obeyed in any kind of analysis:

(i) Vibrational characterization of the interaction.

(ii) Choice of an appropriate theoretical approach to be used in the spectroscopy analysis.

Thus, this current review is guided by these two features, whereby the Quantum Theory of Atoms in Molecules (QTAIM)^204–210 is used to interpret accurately the red- and blue-shifts on the stretch frequencies of the proton donors upon the formation of intermolecular interactions formed by lone pairs of electrons^211–214 or π clouds as proton acceptors,²¹⁵ dihydrogen bonds,²¹⁶ and other similar interactions.^217–223 In this view, once again it is to be hoped that all the concepts and conclusions brought together here can be useful for further studies not only of the aforementioned intermolecular interactions, but other types should be also included.^224–227

The wide chemical range of the π⋯H hydrogen bonds and H^−δ⋯H^+δ dihydrogen bonds

As reported by Kim and collaborators,^228–230 both hydrogen bonds and dihydrogen bonds are widely studied and their properties have been clearly unveiled. Regarding to π⋯H hydrogen bonds, in particular, it is worth mentioning that the p clouds are accessible from the spectroscopy viewpoint. It is worth mentioning that the π clouds are accessible from the spectroscopy viewpoint, particularly as a result of their ability to form conjugated systems and chromospheres.²³¹ In the case of dihydrogen bonds, the interaction between the hydride and the boron or B–H groups provides important insights into interactions in the solid state of matter.²³² In a review elaborated by Singh and co-authors,²³³ it was demonstrated that not only classic hydrogen bonds and ionic interactions,²³⁴ but also π⋯H hydrogen bonds are relevant for the design of nanomaterials, such as those formed by fullerenes and ionospheres. The π⋯H hydrogen bonds are currently characterized in nanoscience by their weak interaction strengths, where the contribution of the dispersion energy is dominant upon the formation and stabilization of the analyzed system.²³⁵ Regarding the chemistry of materials, the dihydrogen bond function also has been established. In a recent paper, Chu et al.²³⁶ reported an alternative procedure for hydrogen storage widely useful in fuel cell technologies. Similar to dihydrogen bonds where alkaline earth elements or metals bind with hydrogen, the interaction observed was H^−δ⋯Ca⁺² in a series of calcium borohydride diammoniates. Furthermore, the properties of alkali-metal aminoborane have been also effectively used for hydrogen storage in a proposal called ‘hydrogen economy’, but with a typical dihydrogen bond between LiH^−δ237,238 and H^+δNH₂BH₃.²³⁹

π⋯H hydrogen bonds

Since the first publications of Pauling^85,240–241 until some more recent works,^242,243 the electronegativity is still being considered a reliant parameter to investigate the formation of atomic interactions.²⁴⁴ In more recent years, elements with higher electronegativity than hydrogen have been used in the proton acceptor framework.^245,246 In addition to the three atoms N, O, and F,²⁴⁷ recent theoretical studies have shown that some regions containing high levels of electron density are potent proton acceptors²⁴⁸ such as the C=C and C≡C bonds. As is widely known, hydrogen-bonded complexes formed by π bonds and proton donors are thus stabilized by the formation of π⋯H hydrogen bonds.^249–257 However, this statement was not entirely accepted at first, and thereby various works were elaborated in order to discover whether individual atoms or π-electrons can be really considered protons acceptors. In this scenery the neutron diffraction analysis has shown that π⋯H interactions are aligned directly at the midpoint of the π bond.²⁵⁸ Indeed, this phenomenon has also been studied theoretically by means of ab initio^259–262 and Density Functional Theory (DFT) calculations^263–267 and also observed through the analysis of rotational^268,269 and infrared spectra,^270,271 as well as by the association of experimental procedures and theoretical calculations compatible to those abovementioned.²⁷² In the electrophilic addition reaction of hydrofluoric acid to the C≡C bond of the acetylene with formation of vinylfluoride (4), an important intermediary is identified as the hydrogen-bonded complex C₂H₂⋯HF,²⁷³ which is formed by the π charge transfer toward the σ* anti-bonding orbital of the hydrogen on the hydrofluoric acid. The hydrogen complex C₂H₂⋯HF (2) is characterized by a T-shape structure with C_2v symmetry wherein the π⋯H interaction is linked at the midpoint of the C≡C bond of the acetylene,^274,275 as can be seen in Fig. 1. In opposition to the conformation analysis for the hydrogen bond symmetry proposed by Perrin,²⁷⁶ note that the main focus here is dedicated just to structural evidence.

Fig. 1 Electrophilic addition reaction of one hydrofluoric acid to the C≡C bond of the acetylene and the formation of the C₂H₂⋯HF hydrogen-bonded complex (2).

Nevertheless, according to studies documented by van der Veken et al.^277,278 concerning the π hydrogen complexes formed by ethylene and cyclopropane as proton acceptors, non-elementary reactions composed by more than two systems are favoured because the second acid molecule improves the intermolecular stability.^279,280 In fact, the reaction mechanism between ethylene and hydrofluoric acid was divulged on the basis of a structural analysis of the complex C₂H₄⋯2(HF), in which the double acid addition is energetically favoured by 38 kJ mol⁻¹ in comparison with systems formed with only one acid as proton donor.²⁸¹ A ternary hydrogen-bonded complex (2′) would thus also be formed by the hydrofluoric acid and acetylene, as predicted in Fig. 2. As for isolated species, it is often argued that their molecular properties are significantly affected by the formation of hydrogen complexes.^{120,282–286} Indeed, several studies have exploited the molecular identities of the hydrogen-bonded complexes C₂H₂⋯HF and C₂H₂⋯2(HF) by analyzing structural arrangements, electronic data, and spectroscopy parameters,^287,288 but the application of a methodology capable of describing the charge density distribution and thereby characterizing the π⋯H interaction²⁸⁹ should be worthwhile. To the best of our knowledge, according to Grabowski²⁹⁰ and Falvello²⁹¹ the ‘different faces of hydrogen bonding’ and ‘the hydrogen bond, front and center’ are great points of view that should be taken into account. Ideally, it is prudent to presume that the interaction between the hydrocarbon ion and its neutral component also may be resultant of electrostatic interactions and charge transference. With the progress in the study of intermolecular interactions, a lot of researches have been conducted in order to elucidate the nature of the proton systems formed exclusively of hydrocarbons, as already highlighted and revised by Meot-Ner.²⁹² The main aim of such researches is to develop a theoretical investigation of systems composed of neutral and ionic hydrocarbons, through which the interaction types ‘cation⋯π’^293–298 and ‘anion⋯π’^299–304 are formed.

Fig. 2 Electrophilic addition reaction of two hydrofluoric acid molecules to the C≡C bond of the acetylene and the formation of the C₂H₂⋯2(HF) hydrogen-bonded complex (2′).

In spite of the large number of hydrocarbons existent in nature, some researches were developed on a model system whose chemical entity and properties are well known: the ethyl cation (C₂H₅⁺) (7). This cationic hydrocarbon is frequently chosen for study because its electron structure displays an interesting phenomenon so-called the hyperconjugation effect³⁰⁵ (see Fig. 3). By definition, the hyperconjugation effect of the ethyl cation occurs when the non-localized hydrogen H⁺ interacts with the carbons of the ethane. This H⁺ proton donor can be used to demonstrate the interaction ability of the ethyl cation towards the π bonds of hydrocarbons,^306,307e.g., the triple (C≡C), double (C=C), and σ (C–C) bonds of the acetylene (C₂H₂), ethene (C₂H₄), and cyclopropane (C₃H₆), respectively. In this context, an objective was projected to confirm whether the hyperconjugation effect of the ethyl cation continues after the formation of the C₂H₅⁺⋯C₂H₂, C₂H₅⁺⋯C₂H₄, and C₂H₅⁺⋯C₃H₆ hydrogen complexes. Finally, an intimate comparison between the molecular properties of these systems and those whose electronic sites are formed by pseudo-π³⁰⁸ and π³⁰⁹ bonds must be worthwhile.

Fig. 3 Schematic drawing of the electronic delocalization on the ethyl cation. Hyperconjugation effect can be visualized in (7).

H^+δ⋯H^−δ dihydrogen bonds

Even taking into account modern studies of intermolecular interactions based on the electronegative criterion,^310,311 amazingly other ones highlight uncommon hydrogen interactions formed by elements (M) with higher electropositivity bound to hydrogen (H), by which an ambiguous character to this atom is revealed.³¹² It is through the formation of M^+δ–H^−δ bond and the improved ‘electronegative power’ of the hydrogen that the so-called dihydrogen bonds (M^+δ–H^−δ⋯H^+δ) are formed.³¹³ In practice, M^+δ–H^−δ need not be the result of a dihydrogen bond, can be a metal–hydrogen bond if M is an alkaline earth or else a transition metal,³¹⁴ for instance. Nevertheless, after the dihydrogen bonds were discovered,¹⁷⁶ intermolecular chemistry has expanded to new horizons, such as studies of solid state, catalysis, crystal engineering, and materials chemistry.^315–317 Among these, many other works about dihydrogen bonds^180,318,319 are conducted by means of theoretical analysis in small systems,^320–322 generally those where M is beryllium, aluminium, gallium, silicon, or germanium.^323a However, the literature reports that beryllium hydride (BeH₂) is the most efficient proton acceptor to be used in the formation of dihydrogen-bonded systems.^323b From a structural viewpoint, the Be–H bond length is slightly increased upon the formation of the BeH₂⋯HX complex with X = F, Cl, CN, and CCH, although the H–X distance is drastically elongated when the strength of the dihydrogen bond H^−δ⋯H^+δ is taken into account.³²⁴ This is widely established for traditional hydrogen-bonded systems formed by the Y⋯H^+δ interaction.³²⁵

Vibrational red- and blue-shifts in intermolecular interactions

The applicability of spectroscopy techniques to interpret bound systems is greatly useful and constantly utilized by the most renowned research groups all over the world.^326–344 This great importance occurs due to the constant eminence of discovering hydrogen bond evidences on the most diversified systems.^345–349 In the specific case of the formation of traditional hydrogen-bonded complexes, the variations in charge density provoke the appearance of red- and blue-shifts in the infrared spectrum often related to the proton donors.^350–357 Hitherto, the quantification of the charge transfer is fundamental in this regard^358–364 because these vibrational phenomena are immediate consequences of the drastic structural deformations in both proton donors and acceptors following complexation.^365–367 One interesting example of deformation on monoprotic acids is related to their bond lengths,^368–369 whereby the red-shift effect is observed when the stretch frequencies are shifted to the lowest values followed by large increases in the absorption intensities^370,371 and thereby the formation of red-shifting hydrogen bonds are manifested.

In regards to the blue-shift effect, this vibrational event was detected firstly by Pinchas,³⁷² Adcock and Zhang,³⁷³ and Sandorfy et al.³⁷⁴ Some period later, however, Hobza et al.^375–377 have studied this blue-shift effect in the context of the intermolecular interactions, which were then named as blue-shifting hydrogen bonds. By definition, the strengthening of the bonds of the proton donors is characteristic of blue-shifting hydrogen bonds because their stretching frequencies are displaced to upward values in the vibrational infrared spectrum.^378,379 Although it is very common to interpret blue-shifts in specific proton donors, such as those found in haloform complexes,³⁸⁰ weakly bound complexes with π proton acceptors,³⁸¹ amino acid fragments,³⁸² or even the C–H bond of the fluoroform (HCF₃)³⁸³ by means of the formation of the C₂H₄O⋯HCF₃ complex, it is also interesting to admit the possibility to form intermolecular π complexes between fluoroform and ethylene (C₂H₄), propylene (C₂H₃(CH₃)), and t-butylene (C₂H₂(CH₃)₂).³⁸⁴ Instead, by taking into account the hydrides BeH₂, LiH, MgH₂, and NaH, as well as HCF₃, it is also important to know whether the molecular properties of these systems are drastically affected upon the formation of dihydrogen-bonded complexes (alkaline interactions)³⁸⁵ and blue-shifting systems (fluoroform as a proton donor),³⁸⁶ respectively. In light of this, a theoretical investigation into the possibility of these monomers forming intermolecular complexes, such as BeH₂⋯HCF₃, LiH⋯HCF₃, MgH₂⋯HCF₃, and NaH⋯HCF₃ was documented.³⁸⁷ Of course, it is already established that these systems are formed by the dihydrogen bonds H^−δ⋯H^+δ, but it is worthwhile to be aware whether the blue-shift of the C–H bond of the HCF₃ can be identified by the analysis of the harmonic infrared spectrum.^388–391

Furthermore, it is not obligatory that intermolecular complexes should be formed through only one hydrogen bond.^392–395 Bifurcate hydrogen bonds or even those possessing multiple sites with donors (Lewis acid)³⁹⁶ and acceptors (lone-electron pairs or π clouds) of protons can lead to the formation of ternary systems.^{350,371,378,397} Thus, it is also important to investigate the formation of ternary dihydrogen-bonded complexes,^398,399 such as NaH⋯2(HCF₃) and NaH⋯2(HCCl₃), and a comparative analysis of their properties with those obtained for the binary complexes NaH⋯HCF₃ and NaH⋯HCCl₃³⁸⁷ must be carried out.

The use of the QTAIM topology for intermolecular modelling

On a purely theoretical chemistry basis, there are many methods^400–408 implemented to investigate the electronic structure and its intrinsic phenomena.^409–414 The use of ab initio^415–420 or DFT calculations,^421–434 semiempirical methods,^435–440 molecular dynamics simulations^441–445 or even the discrete analysis of the solvent effect,^446–451 the nature of the chemical bond, in particular the hydrogen bond, still is an essentially hazy question.^452–454 Surely, the appearance of quantum mechanics has helped many research groups to develop quantum chemical methods,^455,456 principally the QTAIM formalism designed by Bader in the 1960s.^457–459 In a brief comment, Bader adopted the pure principles of quantum mechanics, whose great objective was the description of atomic behaviour within the molecular environment,^460–462i.e., molecular fragments in chemical bonds.^463,464 Moreover, Bader's insight was also based on Daltonian atomic theory,^465,466 according to which the molecular structure is shaped into a cooperative ensemble.⁴⁶⁷ Thus, Bader has defined cooperative atoms in a molecule as open systems capable of exchanging charge and momentum with their neighbours, meaning that atoms can be treated as quantum mechanics entities within the molecules.⁴⁶⁸ Through the historical background of the QTAIM, several novel approaches have been idealized, such as the Two-Component Quantum Theory of Atoms in Molecules (TC-QTAIM),⁴⁶⁹ Quantum Division Basins (QDB),⁴⁷⁰ and Quantum Theory of Atoms in Positronic Molecules (QTAIPM),⁴⁷¹ for instance.

Nevertheless, the QTAIM architecture consists of mathematical conceptions derived from the theories of Dirac, Feynman, and Schwinger,^472–474 whose contributions give the foundation for the principle of the least action for particle motion. In classical terms, the principle of least action states that a quantity (q) derived from the wave function is minimized in space and time (t₁ to t₂):

where L̄ is the Lagrangian differential operator defined by the contributions of the kinetic (K) and potential (U) energies as follows: L = K − U. In a review containing the basic principles of QTAIM, Bader⁴⁷⁵ and Nasertayoob and Shahbazian⁴⁷⁶ have refined some topological concepts. For instance, the atomic frontier of an open system is defined as a zero-flux surface, whereby:

(i) The time variation at the end point is zero.

(ii) The surface is zero when the functions are at the extreme.

These two arguments provide support for the QTAIM explanation of how atoms bond to form molecules.⁴⁷⁷ By taking over that the two propositions given above are obeyed in concordance with the integration of the eqn (1), by which is obtained:

This equation states that minimization of action occurs if a differential equation is solved for the surface. In other words, δW₁₂ vanishes according to the Euler–Lagrange equation,⁴⁷⁸ and the Schrödinger equation for a normalized wave function can be determined as ĤΨ* − EΨ* = 0 and ĤΨ − EΨ = 0. In his article published in 1926,⁴⁷⁹ Schrödinger concluded that the state function ψ and the Hamiltonian operator Ĥ for the hydrogen atom are related as follows:

Whose expression for the Hamiltonian operator is Ĥ = − ℏ²/2m∇² + Û. There are two forms to express the kinetic energy (K and G), which jointly furnish the Laplacian for one-electron according to eqn (4): Integrating eqn (4) over a spatial region S_(Ω) gives:where K and G represent the kinetic energy densities⁴⁸⁰ which are equivalent to the Laplacian of the charge density, (∇²ρ).^481,482 If the zero-flux is one at any point of the surface S_(Ω) where n is a normal vector,⁴⁸³ then K = G and the condition to be satisfied is:

∇ρn = 0 for all points on the surface S_(Ω) (6)

This equation defines the surface by which the atom is delimited as zero-flux charge density.^484–486 For an ordinary quantum system, it is necessary to impose the vanishing of the wave functions ψ and ψ* at the quantum boundary. Well, according to the Born postulates,⁴⁸⁷ the product ψψ*dτ represents a probability distribution (eqn (7)), which, on the surface, should be zero.

ρ = N∫ψ*ψdτ (7)

The relation between surface conditions and high and low electron density sites is governed by the virial theorem.^488,489 By considering the contributions of kinetic and potential energies, molecular regions with reduction and increase of charge density are represented by the positive (kinetic energy density G is positive) and negative (potential energy density U is negative) values of the Laplacian^490,491 as demonstrated by eqn (8) and (9):

or

In general, this short overview shows that QTAIM^492–500 is a useful approach to the study of chemical bonds and intermolecular interactions, and furthermore, it identifies Bond Critical Points (BCP)⁵⁰¹ between interactive atoms^502–506 through the calculations of topological parameters related to the molecular stability, from which the electron density profile and its Laplacian field are extracted.⁵⁰⁷ Based on the kinetic and potential energy operators given above,^508,509 the QTAIM theory parameters identify maxima and minima of the electron density on the surface of the molecule and classify the chemical bonds as closed shells guided by ∇²ρ > 0, or additionally as shared interactions if ∇²ρ < 0.^510,511 This theoretical argument has been used successfully to establish the intermolecular topology of several systems,^512–531e.g., compounds with π bonds⁵³² and binary complexes,^533,534 including those formed by acetylene and hydrofluoric acid.⁵³⁵ It is not common to study ternary hydrogen complexes from the theoretical viewpoint.²⁸¹ However, even if it can be done, the QTAIM results should be derived either from the equilibrium geometry⁵³⁶ or from fitting parameters derived from the energy.⁵³⁷

Likewise for Rozas et al.,⁵³⁸ the inherent qualities of the QTAIM warrants its application in the identification of the interactions in the C₂H₂⋯2(HF) complex, but it should be remembered that the result of the interaction between the C≡C bond of the acetylene and two hydrofluoric acid molecules reveals an unusual π hydrogen bond.⁵³⁹ The parameters related to the C₂H₂⋯HF binary complex thus represent a starting point to discuss the formation of π⋯H, not only in the C₂H₂⋯2(HF) ternary complex, but also in other systems formed by novel interactions.^540–547 Besides the π⋯H hydrogen bonds, the QTAIM method has also been very useful for studying H^−δ⋯H^+δ dihydrogen bonds. For example, Popelier^186a and later Grabowski and Leszczynski⁵⁴⁸ have carried out theoretical studies of systems stabilized by dihydrogen bonds by means of ab initio calculations and QTAIM integrations. As it is widely known that suitable analysis of vibrational modes is a decisive factor for describing adequately the formation of intermolecular interactions,^549–551 our group reported a theoretical study of dihydrogen-bonded complexes formed by BeH₂⋯HX with X = F, Cl, CN, and CCH, in which was proposed the investigation not only of their structural and electronic parameters, but also the characterization of their stretch frequencies and red-shift effects.^552,553

According to Caballero and Jalón,⁵⁵⁴ the ‘electronegative capacity’ of H^−δ in BeH₂ can be demonstrated by a theoretical analysis of the ternary dihydrogen-bonded complex, BeH₂⋯2(HCl).⁵⁵⁵ This observation led us to invest in a computational study in order to confirm the capability of HCN and HNC to form this ternary dihydrogen-bonded complex, not only by evaluating the terminal hydride (8) as a proton acceptor in BeH₂⋯2(HX), but also simultaneously two (9) hydrides in XH⋯BeH₂⋯HX, in this case X = CN and NC, as illustrated in the Fig. 4.⁵⁵⁶ Ideally, the characterization of intermolecular interactions, in particular dihydrogen bonds formed by multiple sites either of donors or acceptors of protons or any other ones can be performed by computing the molecular charge density, according to which the explanation of the cooperative effect related to molecular energy becomes possible^557,558 besides the elucidation of the dynamism of the infrared vibrational modes, e.g., the redshifts on the stretch frequencies of the proton donors.^559–566 The questions regarding the red- and blue-shifting hydrogen bonds are debated constantly in the literature.^567–574 One of the most important topics is the presentation of new theoretical approaches, such as the analysis of the induced dipole proposed by Hermansson⁵⁷⁵ for investigating the origin of the vibrational blue shift. In line with this, the magnitude and direction of derivatives related to the dipole enhancements and bond lengths^576,577 corroborate and reinforce the vibrational changes, either red or blue, in the proton donors.⁵⁷⁸ According to some studies,^579–581 the applicability of this electropotential method has been proven. However, the main aim of this review is to demonstrate the QTAIM capability to interpret the vibrational red- and blue-shift events after the formation of hydrogen bonds (π⋯H) and dihydrogen bonds (H^−δ⋯H^+δ) by the quantification of their electronic densities (ρ) and Laplacians (∇²ρ), as well as by the whole topological analysis.^582–585

Fig. 4 Illustration of the BeH₂⋯HX bifurcate (8) and XH⋯BeH₂⋯HX linear (9) ternary dihydrogen systems (X=CN and NC).

The π⋯H hydrogen bonds: structure and vibrational spectrum

It should be pointed out that exploring theoretically the minimum of the potential energy surface,⁵⁸⁶ the optimized geometries of π hydrogen complexes were determined through the execution of the B3LYP/6-311++G(d,p) calculations. By this insight, the structures of the hydrogen complexes C₂H₂⋯HF (10), C₂H₂⋯2(HF) (10′), C₂H₄⋯HF (11), and C₂H₄⋯2(HF) (11′) obtained by this B3LYP/6-311++G(d,p) quantum level are shown in Fig. 5 (acetylenic complexes) and Fig. 6 (ethylenic complexes). Some studies reported by the specialized literature^587–590 have shown that the formation of these complexes can be interpreted by the charge transfer⁵⁹¹ flux from the bonding orbital (π) towards the antibonding orbital (σ*) of the unsaturated bonds (C=C and C≡C) and proton donors (HF, HCl, HCN, or HCCH),^273,538 respectively. Thereby, a series of drastic structural deformations on the molecular subunits are consequently observed, such as the weakness on the C¹C², H^c–F^d, and H^e–F^f bonds, whose values are listed in Table 1. Once this charge transfer phenomenon is taken into account, it can be clearly seen that, in almost all the complexes, C₂H₄⋯HF, C₂H₄⋯2(HF), C₂H₂⋯HF, and C₂H₂⋯2(HF), the most affected bonds are C≡C and H–F, while the C–H bonds remain almost unchanged.

Fig. 5 Geometries of the C₂H₂⋯HF (10) and C₂H₂⋯2(HF) (10′) hydrogen complexes using the B3LYP/6-311++G(d,p) calculations.

Fig. 6 Geometries of the C₂H₄⋯HF (11) and C₂H₄⋯2(HF) (11′) hydrogen complexes using the B3LYP/6-311++G(d,p) calculations.

Table 1 Values of the structural enhancements (Δr), hydrogen bond distances (R), and linearity deviation (θ) for the 2(HF), C₂H₂⋯HF (10), C₂H₂⋯2(HF) (10′), C₂H₄⋯HF (11), and C₂H₄⋯2(HF) (11′) complexes obtained from B3LYP/6-311++G(d,p) calculations^a

Parameters	Hydrogen complexes
	2(HF)	(10)	(10′)	(11)	(11′)
a Values of Δr and R are given in Å. Values of θ are given in degrees.
ΔrC^1–H^a	—	0.001	0.001	−0.000(5)	0.003
ΔrC^1–H^a′	—	—	—	−0.000(5)	0.003
ΔrC^2–H^b	—	0.001	0.004	−0.000(5)	0.003
ΔrC^2–H^b′	—	—	—	−0.000(5)	0.003
ΔrC^1=C^2	—	—	—	0.004	0.008
ΔrC^1≡C^2	—	0.002	0.002	—	—
ΔrH^c–F^d	0.003	0.009	0.020	0.011	0.024
ΔrH^e–F^f	0.007	—	0.012	—	0.018
R(H^c⋯C^1)	—	2.234	2.176	2.273	2.141
R(H^c⋯C^2)	—	2.234	2.105	2.273	2.141
R(F^d⋯H^e)	1.831	—	1.777	—	1.680
R(F^f⋯H^b)	—	—	2.547	—	2.458
θ^1	—	0.65	1.45	116.70	117.80
θ^2	—	0.65	1.93	116.70	117.80
θ^3	—	—	32.10	—	27.80
θ^4	12.10	—	24.10	—	22.30
R(π⋯H)	—	2.151	—	2.173	2.034
R(p-π⋯H)	—	—	—	—	—
R(un-π⋯H)	—	—	2.105	—	—

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As justified by Del Bene,⁵³⁹ the π hydrogen bond in the C₂H₂⋯HF complex is formed exactly at the midpoint of the C≡C bond of the acetylene, whose distance value is 2.151 Å. The T-shape geometry of the C₂H₂⋯HF complex can thus be fully described by the C_2v symmetry because the distances between H^c and both C¹ and C² (H^c⋯C¹ and H^c⋯C²) are precisely equal to 2.234 Å.⁵⁹² In the C₂H₂⋯2(HF) complex, however, the dimer of the hydrofluoric acid forms one hydrogen bond which is not aligned in the midpoint of the C≡C bond of the acetylene. This can be clearly seen due to the distances of R_(H^c⋯C¹) and R_(H^c⋯C²), and precisely by their correspondingly distinct values of 2.176 and 2.105 Å the formation of one asymmetrical hydrogen bond profile is favoured.⁵⁹³ Indeed, the interaction R_(H^c⋯C²) is stronger once its length is shorter, as highlighted by Parra and Zeng⁵⁹⁴ as well as by Remer and Jensen,⁵⁹⁵ and thereby indicates an interaction between the H^c hydrogen atom of the hydrofluoric acid and the C² carbon of the acetylene molecule in the C₂H₂⋯2(HF) ternary system. Despite the π hydrogen bond aforementioned, a similar result of shortening of hydrogen bond in halothane:ammonia complexes has been reported by Michielsen, Dom, van der Veken, Herrebout et al.⁵⁹⁶ In the C₂H₂⋯HF binary, the charge transfer from the C≡C bond of the acetylene to the HF molecule causes elongation of the H^c–F^d bond. This is also confirmed by the 2(HF) dimer along the structure of C₂H₂⋯2(HF), where the shorter distance of the hydrogen bond F^d⋯H^e provokes enlargement in the H–F bond lengths as follows: r_(H^c–F^d) > r_(H^e–F^f). In spite of this, some important structural modifications in 2(HF) also have been observed, such as the contraction from 1.831 to 1.777 Å related to the distance of R_(F^d⋯H^e).

In acetylene, the distance between C² and H^b is longer than the distance between C¹ and H^a. This occurs due to a possible interaction between the H^b hydrogen of the acetylene and the F^f fluorine of the second molecule of hydrofluoric acid (H^e–F^f). Thus, the F^f⋯H^b interaction is likely to exist, because its distance of 2.547 Å corroborates the values of the van der Waals radii of hydrogen (H = 1.2 Å) and fluorine (F = 1.35 Å), which together sum 2.55 Å.^597–602 Another important indication of a possible interaction between H^b and F^f are the variations in the bond angles θ¹, θ², and θ⁴. It should be noted that θ¹ = θ² in the C₂H₂⋯HF complex, while, in the C₂H₂⋯2(HF) hydrogen complex, θ¹ > θ². Furthermore, the θ⁴ angle in 2(HF) undergoes drastic enlargement owing to the formation of the C₂H₄⋯2(HF) and C₂H₂⋯2(HF) hydrogen-bonded complexes, which varies from 12.10 up to 22.30 and 24.10°, respectively. The θ⁴ angle deviation is yet another structural confirmation that R_(F^f⋯H^b) exists in the C₂H₄⋯2(HF) and C₂H₂⋯2(HF) complexes.⁶⁰³

The main spectroscopy event in the structures of hydrogen complexes is the displacement in the stretch frequencies of the proton donors to downward energy regions of the vibrational spectrum, an effect widely known as red-shift.^274,604,605 So, the red-shifts of the C₂H₂⋯HF and C₂H₂⋯2(HF) complexes were computed at the B3LYP/6-311++G(d,p) level of theory, whose values are listed in Table 2. The formation of the dimer 2(HF) confirms that the stretch frequencies of H^c–F^d and H^e–F^f are shifted in −37.9 and −138.8 cm⁻¹ to downward values, which are in good agreement with the respective experimental data of −31.0 and −93.0 cm⁻¹.^273a,b,369 Likewise, the greater increase in infrared intensity can also be found in the H^e–F^f bond, whose ratio is 3.7, whereas that in H^c–F^d is only 1.1. Nevertheless, the red-shift of −31 cm⁻¹ on the stretch frequency in 2(HF) is shorter than the respective value for C₂H₂⋯HF, which in turn, is also shorter in comparison with the C₂H₂⋯2(HF) complex. In other words, the value is −438.0 cm⁻¹ in C₂H₂⋯2(HF), whereas in C₂H₂⋯HF is −231.0 cm⁻¹, which accords with the available experimental value of −215.0 cm⁻¹. On the other hand, it can also be seen in Fig. 7 that, in the C₂H₂⋯2(HF) complex, is displaced much more than . The formation of the C₂H₄⋯2(HF) and C₂H₄⋯HF hydrogen-bonded systems provokes the arising of a more accentuated red-shift on the ternary ones, in the region of −463.1 and −259.6 cm⁻¹, respectively, as explained above for C₂H₂⋯2(HF) and C₂H₂⋯HF.

Table 2 Vibrational shift frequencies (Δν^Str_HF), intensities ratios for the hydrofluoric acid bonds after complexation (I^Str_HF,c/I^Str_HF,m), stretching frequencies of the hydrogen bonds (ν^Str_R), and absorption intensities (I^Str_R) for the 2(HF), C₂H₂⋯HF (10), C₂H₂⋯2(HF) (10′), C₂H₄⋯HF (11), and C₂H₄⋯2(HF) (11′) complexes obtained at the B3LYP/6-311++G(d,p) level of theory. Experimental data in parentheses^c

Parameters	Hydrogen complexes
	2(HF)	(10)	(10′)	(11)	(11′)
a Ref. 273a.b Ref. 273b.c (monomers): C2H4 = 1683.9 cm^−1 and C2H2 = 2062.5 cm^−1; (monomers): C2H4 = 0.0 km mol^−1 and C2H2 = 0.0 km mol^−1; ν^StrHF and I^StrHF (monomer): 4096 cm^−1 and 130 km mol^−1 respectively.
	−37.9 (−31)^a	−231.0 (−215)^b	−438	−259.6	−463.1
	1.1	5.6	7.3	6.1	4.7
	−138.8 (−93)^a	—	−246	—	−292.5
	3.7	—	3.8	—	4.3
	—	—	—	−8.5	3.7
	—	—	—	1.3	1.4
	—	−6.1	−12.9	—	—
	—	1.2	1.8	—	—
ν^Str(π⋯H)	—	137.8	—	133	183.8
I^Str(π⋯H)	—	1.0	—	0.95	4.0
ν^Str(un-π⋯H)	—	—	181.5	—	—
I^Str(un-π⋯H)	—	—	3.6	—	—
	—	—	209	—	253.3
	—	—	8.1	—	0.6
	—	—	70.2	—	143.8
	—	—	7.9	—	8.6

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Fig. 7 Simulation of the vibrational harmonic infrared spectrum for the C₂H₂⋯2(HF) hydrogen-bonded bond complex using the B3LYP/6-311++G(d,p) calculations. The displacement vectors indicate the stretches of the new vibrational modes ν^Str_(π⋯H), , and .

The formation of the C₂H₄⋯HF, C₂H₄⋯2(HF), C₂H₂⋯HF, and C₂H₂⋯2(HF) hydrogen complexes gives rise to the new vibrational modes known as hydrogen bond stretch frequencies.⁶⁰⁶ This kind of vibrational mode exhibits very interesting features, such as low stretch frequency and weak absorption intensity.^607,608 In comparison with the π complexes, the hydrogen bond stretch frequencies, ν^Str_(π⋯H) and ν^Str_(un-π⋯H), of the C₂H₄⋯2(HF) and C₂H₂⋯2(HF) hydrogen complexes are strongest, and the values of 183.8 and 181.5 cm⁻¹ are identified in accordance with the directional arrows illustrated in Fig. 7. In concordance with the foregoing results of the analysis of the C₂H₄⋯HF and C₂H₂⋯HF T-shaped complexes, the oscillator ν^Str_(π⋯H) is characterized as a stretch mode at the midpoint in both acetylene and ethylene.⁶⁰⁹ The C₂H₂⋯2(HF) ternary complex is considered a system apart because the stretch frequency of its hydrogen bond does not appears as π⋯H. Amazingly this stretch frequency is identified as unusual (un), un-π⋯H, once its harmonic oscillator is composed by the hydrogen (H^c) of the first hydrofluoric acid (H^c–F^d) and the carbon (C²) of the acetylene, respectively. Moreover, the oscillator of the additional interaction is also observed, although at a much lower frequency, in the region of 70.2 cm⁻¹ with an absorption intensity of 7.9 km mol⁻¹. Besides the new vibrational modes, the formation of the C₂H₄⋯HF, C₂H₄⋯2(HF), C₂H₂⋯HF, and C₂H₂⋯2(HF) hydrogen complexes exhibits a slight vibrational variation in the C≡C, C=C, and C–H bonds, and therefore these results were not considered.

Fig. 8 illustrates the optimized geometries of the complexes C₂H₄⋯HCF₃ (12), C₂H₃(CH₃)⋯HCF₃ (13), and C₂H₂(CH₃)₂⋯HCF₃ (14) obtained through the B3LYP/6-311++G(d,p) calculations. As depicted in Table 3, one important initial observation was the slight contraction of −0.000(2) and elongation of 0.000(3) Å in the lengths of the C³–H^c bonds of HCF₃ in the 12 and 14 complexes, respectively. In a middle term, C³–H^c in 13 remained unchanged. Investigations of the hydrogen bond distances have shown that 13 and 14 are the respective shortest and longest bonded complexes, whereas 12 is of intermediate length. However, it is not possible to establish any relationship between the interaction strength and the variation on the C³–H^c bond length. Most likely this unsystematic trend is caused by the hydrogen bond profile in 13, where, instead of a contact between the π bond of propylene and the hydrogen of HCF₃, a direct interaction on the carbon can be seen wherein this element is acting as a proton acceptor. The same trends determined for hydrogen bond distance also are applied to their vibrational stretching frequencies, which are 50.5, 58.4, and 47.1 cm⁻¹ for the 12, 13, and 14 complexes, respectively.

Fig. 8 Optimized geometries of the blue-shifting hydrogen-bonded complexes C₂H₄⋯HCF₃ (12), C₂H₃(CH₃)⋯HCF₃ (13), and C₂H₂(CH₃)₂⋯HCF₃ (14) by using the B3LYP/6-311++G(d,p) calculations.

Table 3 Values of the structural parameters and infrared modes of the HCF₃ monomer, as well as of the C₂H₄⋯HCF₃ (12), C₂H₃(CH₃)⋯HCF₃ (13), and C₂H₂(CH₃)₂⋯HCF₃ (14) complexes obtained by using the B3LYP/6-311++G(d,p) calculations^a

Parameters	Hydrogen complexes
	HCF3	(12)	(13)	(14)
a Values of R and r are given in Å. Values of ν^Str and I^Str are given in cm^−1 and km mol^−1 respectively.
R(H^c⋯π)	—	2.878	—	3.120
R(C^2⋯H^c)	—	—	2.801	—
r(C^3–H^c)	1.089(6)	1.089(4)	1.089(6)	1.089(9)
Δr(C^3–H^c)	—	−0.000(2)	0.000	0.000(3)
r(C^1=C^2)	—	1.3300	1.3329	1.3347
		(1.328)	(1.331)	(1.333)
Δr(C^1=C^2)	—	−0.0012	−0.0017	−0.0017
ν^Str(π⋯H)	—	50.5	—	47.1
I^Str(π⋯H)	—	0.1	—	0.1
	—	49.2	58.4	—
	—	0.09	0.5	—
	3141.2	3144.4	3140.7	3136.8
	32.8	4	2.9	0.98
	—	3.2	−0.5	−5.0
	—	0.12	0.08	0.03

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In regards to the structure and spectroscopy of the proton donors,⁶¹⁰ some of them closely similar to those revisited here, an interesting observation is the direct relationship between the variations of their bond lengths, Δr_(C³–H^c), versus the shifting of the stretch frequencies, . It should be noted that the blue-shift of 3.2 cm⁻¹ in 12 is justified by the decrease of −0.0002 Å, and the red-shift of −5.0 cm⁻¹ in 14 can be explained by the slight increase of 0.0003 Å, just as that found by Hippler⁶¹¹ in spectroscopy studies of hydrogen-bonded complexes in the gas phase. The invariant C³–H^c bond in 13 also remains non-shifted in the infrared spectrum, corroborating the structural and vibrational parameters. Nevertheless, the red-shift values of −0.5 (13) and −5.0 cm⁻¹ (14) are followed of a decrease in their absorption intensities. These can be contradictory results, but theoretical and experimental evaluations of Herrebout et al.⁶¹² have shown that not all red-shifts are accompanied by a reduction in the absorption intensities, when π bonds of hydrocarbons bind with trifluoro(halo)methanes, for instance. Of course these spectroscopy profiles may be useful as a guide for experimental investigations,^{369,613–616} once all the results were obtained using the B3LYP functional, which is considered an efficient method for conducting studies of intermolecular interactions⁶¹⁷ and infrared spectra of hydrogen bonds.^618–622

The H^−δ⋯H^+δ dihydrogen bonds: structure and vibrational spectrum

Spectroscopy studies of dihydrogen bonds have attracted much attention in recent years.^623–632 Besides the crystallography as one of the most used techniques in studies of dihydrogen bonds, the contribution of the infrared spectrum is also a powerful tool because there are weak and sensitive stretch frequencies to be unveiled. Here the results of the infrared spectrum of the BeH₂⋯HX dihydrogen complexes (with X = F, Cl, CN, and CCH) are one of the main points of discussion. However, firstly it is essential to discuss of the equilibrium geometries of these systems from the theoretical viewpoint. Fig. 9 exhibits the optimized geometries of the dihydrogen-bonded complexes BeH₂⋯HF (15), BeH₂⋯HCl (16), BeH₂⋯HCN (17), and BeH₂⋯HCCH (18) obtained by using the B3LYP/6-311++G(3d,3p) calculations. In Table 4 are listed the results for the dihydrogen bond distances, R_{(H^−δ⋯H^+δ)}, as well as the increases in the bond lengths, Δr_(Be–H) and Δr_(H–F), related to the beryllium hydride and hydrofluoric acid, respectively. Initially, the interaction strength in the BeH₂⋯HX complexes (with X = F, Cl, CN, and CCH) with regard to the values of the distances R_{(H^−δ⋯H^+δ)} is represented by the following trend: F > Cl > CN > CCH. About BeH₂ and HF, however, Δr_(Be–H) exhibits small variations in comparison to Δr_(H–F). The main results of the infrared spectra analysis for the dihydrogen-bonded complexes BeH₂⋯HX are presented in Table 5. The values of the vibrational mode , understood as stretch frequencies of the dihydrogen bond or new vibrational modes varying between 69.5–173.4 cm⁻¹, are very similar to those observed for the classical hydrogen-bonded complexes.^326–329 Although the vibrational modes must be identified here as stretch frequencies of the dihydrogen bonds, their absorption intensities demonstrate the weakness of the interactions formed in the BeH₂⋯HX complexes, e.g., the values of 0.7 and 0.3 km mol⁻¹ for BeH₂⋯HCN and BeH₂⋯HCCH respectively. However, examination of the red-shift on the stretch frequencies of the Lewis’s acids reveals a synchronism related to the displacement. It can be noted that the longer downward stretch value of −240.0 cm⁻¹ is related to the hydrofluoric acid of the BeH₂⋯HF complex, whereas the shorter result of −15.3 cm⁻¹ is related to the acetylene of the BeH₂⋯HCCH complex.

Fig. 9 Optimized geometries of the BeH₂⋯HX dihydrogen-bonded complexes with X = F (15), Cl (16), CN (17), and CCH (18) using the B3LYP/6-311++G(3d,3p) calculations.

Table 4 Values of the structural parameters for the dihydrogen-bonded complexes BeH₂⋯HX with X = F, Cl, CN, and CCH obtained from the B3LYP/6-311++G(3d,3p) calculations^a

Parameters	Dihydrogen complexes
	X = F (15)	X = Cl (16)	X = CN (17)	X = CCH (18)
a All values are given in Å.
R(H^−δ⋯H^+δ)	1.598	1.792	2.058	2.276
Δr(Be–H)	0.001	0.028	0.003	0.001
Δr(H–X)	0.010	0.010	0.004	0.002

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Table 5 Profiles of the vibrational harmonic spectrum of the dihydrogen-bonded complexes BeH₂⋯HX with X = F, Cl, CN, and CCH by using the B3LYP/6-311++G(3d,3p) calculations^a

Parameters	Dihydrogen complexes
	X = F (15)	X = Cl (16)	X = CN (17)	X = CCH (18)
a ν^Str values are given in cm^−1. I^Str values are given in km mol^−1.
	173.4	115.5	97.7	69.5
	1.9	3.0	0.7	0.3
Δν^StrH–X	−240.0	−148.7	−62.5	−15.3
	6.2	8.9	3.2	2.6

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Similarly to the π⋯H hydrogen bonds mentioned previously, such as those with three centers⁶⁰³commonly known as ternary systems, it is very interesting that dihydrogen bonds can be formed in this same context, i.e., ternary dihydrogen systems.^633,634 Thus, the formation of the BeH₂⋯2(HCl) ternary dihydrogen-bonded complex is related to the possibility of forming the three isomers 19, 20, and 21, whose optimized geometries, accompanied by the BeH₂⋯HCl binary complex 22 illustrated in Fig. 10, are based on the scheme outlined in Fig. 4. Exhibiting a symmetric geometry, the bond lengths of the hydrochloric acid are identical upon the formation of the isomers 20 and 21. For the isomer 19, all its bond lengths are dissimilar, especially the values of 1.741 and 3.262 Å attributed to R_{(H^−δ⋯H^+δ)} and R_(Cl⋯H^+δ), respectively. Moreover, the structure of isomer 19 is shaped by both the dihydrogen bonds _{(H^−δ⋯H^+δ)}-(19)-A and hydrogen bonds _{(Cl^−δ⋯H^+δ)}-(19)-B, suggesting the formation of a mixed interaction system: a dihydrogen-bonded and hydrogen-bonded complex.^635,636 The values of the main vibrational modes of the dihydrogen complexes 19, 20, and 21 are listed in Table 6. It can be seen that the hydrochloric acids have their stretch frequencies (ν^Str) shifted downwards (Δν^Str) and their absorption intensities (I^Str_C) are substantially increased (I^Str_C/I^Str_m), and it is due to these phenomena that the red-shift effects arise.⁶³⁶ It should be noted that the 20 and 21 undergo similar and slighter modifications in Δν^Str_(H–Cl), while Δν^Str_(H–Cl)-(22) and Δν^Str_(H–Cl)-(19)-A are the greatest variations, suggesting that the 19 isomer tends to form a binary complex instead of ternary. In other words, the stretch frequency of the proton donor _(H–Cl)-(19)-B is slightly affected by the formation of BeH₂⋯2(HCl). Thereby, the weakest interaction is formed between the BeH₂⋯HCl binary complex and the extra molecule of hydrochloric acid, _(H–Cl)-(19)-B. The results of the intermolecular stretch frequencies presented in Table 6 demonstrate a low value of 13.8 cm⁻¹ followed by a lowest intensity of 0.1 km mol⁻¹ for _(Cl⋯H^+δ)-(19)-B, which means that the vibrational detection of the dihydrogen bond in the infrared spectrum is not an easy task, as can be seen in Fig. 11.

Fig. 10 Optimized geometries of BeH₂HCl binary dihydrogen-bonded complex (22), as well as ternary dihydrogen-bonded isomers (19), (20), and (21) obtained from B3LYP/6-31++G(3d,3p) calculations. The H–Be and H–Cl bond lengths on free monomers are 1.324 and 1.2807 Å respectively.

Table 6 Values of the stretch frequencies for the BeH₂⋯HCl binary dihydrogen complex (22) and ternaries dihydrogen complexes (19), (20), and (21) obtained at the B3LYP/6-31++G(3d,3p) theoretical level^a

Modes	Vibrational parameters
	ν^Str	Δν^Str	I^StrC	I^StrC/I^Strm
a Values of ν^Str and I^Str are given in cm^−1 and km mol^−1 respectively. Values of ν^StrHCl and I^StrHCl (monomer) 2943 cm^−1 and 44.9 km mol^−1 obtained by using the B3LYP/6-31++G(3d,3p) calculations.
(H–Cl)-(22)	2796	−146.3	392	8.7
(H–Cl)-(19)-A	2764	−179.0	514	11.4
(H–Cl)-(19)-B	2939	−4.0	73.8	1.6
(H–Cl)-(20)-A	2842	−100.0	408	9.0
(H–Cl)-(20)-B	2842	−100.0	408	9.0
(H–Cl)-(21)-A	2821	−122.0	698	15.5
(H–Cl)-(21)-B	2821	−122.0	698	15.5
(H^−δ⋯H^+δ)-(22)	116.1	—	3.0	—
(H^−δ⋯H^+δ)-(19)-A	125.6	—	4.0	—
(Cl⋯H^+δ)-(19)-B	13.8	—	0.1	—
(H^−δ⋯H^+δ)-(20)-A	74.8	—	0.6	—
(H^−δ⋯H^+δ)-(20)-B	74.8	—	0.6	—
(H^−δ⋯H^+δ)-(21)-A	52.4	—	0.0	—
(H^−δ⋯H^+δ)-(21)-B	52.4	—	0.0	—

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Fig. 11 Illustration of the infrared spectrum of the (19) isomer of the BeH₂⋯2HCl ternary dihydrogen-bonded complex using the B3LYP/6-31++G(3d,3p) theoretical level.

By assuming the possibility of the formation of ternary dihydrogen complexes due to the attack of two proton donors on the beryllium hydride, the optimized geometries of the BeH₂⋯2(HCN) (23) and BeH₂⋯2(HNC) (24) bifurcate dihydrogen complexes as well as the HCN⋯BeH₂⋯HCN (25) and CNH⋯BeH₂⋯HNC (26) linear ones are illustrated in Fig. 12 and 13, respectively. First, it can be seen that the bifurcate complexes present a difference in their dihydrogen bond distances, e.g., 2.340 and 2.322 Å, when HCN (A) and HCN (B) interact with BeH₂ in 23. On the other hand, the dihydrogen bond distances are practically identical in the linear complexes, as demonstrated by the results of 1.859 and 1.857 Å for the 25 complex. With regard to H–C and H–N bond lengths, there are not significant differences between the acids (A and B) in all ternary complexes. However, the distortions in the structure of BeH₂ have yielded interesting results, which should be carefully investigated. In the bifurcate dihydrogen complexes, it should be noted that the value of 1.325 Å for the Be–H bond length is decreased to 1.316 Å (^−δH–Be) as well as increased to 1.338 Å (Be–H^−δ) after the formation of 24. Because the formation of the dihydrogen bonds in linear complexes occurs simultaneously on H^−δ, there is a slight reduction on their bond lengths ^−δH–Be and Be–H^−δ, whose respective values for the complexes 25 and 26 are 1.323 and 1.322 Å.

Fig. 12 Optimized geometries of bifurcate (23–24) dihydrogen-bonded complexes BeH₂⋯2HX (with X = CN and NC) obtained by using the B3LYP/6-31++G(3d,3p) calculations. H–Be (BeH₂), H–C (HCN) and H–N (HNC) bond lengths on free monomers are 1.3255, 1.0672, and 1.0007 Å, respectively.

Fig. 13 Optimized geometries of linear (25–26) dihydrogen-bonded complexes XH⋯BeH₂⋯HX (with X = CN and NC obtained by using the B3LYP/6-31++G(3d,3p) calculations. H–Be (BeH₂), H–C (HCN) and H–N (HNC) bond lengths on free monomers are 1.3255 Å, 1.0672 Å and 1.0007 Å, respectively.

The spectroscopy techniques and DFT calculations have been used conjointly for investigating the vibrational modes of bifurcate dihydrogen bonds,⁶³⁷ as reported in the pioneering study conducted by Nikonov et al.⁶³⁸ The values for the main stretching frequencies and absorption intensities of the bifurcate dihydrogen complexes BeH₂⋯2(HCN) (23) and BeH₂⋯2(HNC) (24), and linear ones HCN⋯BeH₂⋯HCN (25) and CNH⋯BeH₂⋯HNC (26) are listed in Table 7. In agreement with other studies,⁶³⁹ it can be seen that the stretch frequencies of the bonds _(H^+δ–C)-A and _(H^+δ–C)-B are displaced downward in value. However, the H^+δ–N bonds of 24 (bifurcate) and 26 (linear) provide the largest red-shifts, with values of −37.4 and −75.3 cm⁻¹ respectively. In bifurcate complexes, red-shifts of −8.5 and −16.9 cm⁻¹ have been found for the bond (Be–H^−δ) of the 23 and 24 complexes. On the other hand, blue-shifts of +23.6 and +45.8 cm⁻¹ have been computed for the bond Be–H^−δ of the 25 and 26 complexes.^640,641 Briefly, the vibrational blue-shifts of (Be–H^−δ) are caused by the shortening of its bond length. The absorption intensity (I^Str_m) of the bond Be–H^−δ in the BeH₂ monomer is zero, although the complexes 23 and 24 have furnished the most evident ratios (I^Str_c/I^Str_m), whose values are 43.9 and 90.0 respectively. Regardless the geometric shape, either linear or bifurcate, the red- and blue-shifts on the stretch frequencies of the bond (Be–H^−δ) were observed in 23–24 and 25–26, indistinctly of whether they are the longest and shortest bonded complexes, respectively.

Table 7 Values of the stretch frequencies of the BeH₂⋯2HX and XH⋯BeH₂⋯HX ternary dihydrogen complexes obtained using the B3LYP/6-31++G(3d,3p) theoretical level^a

Modes	Vibrational parameters
	ν^Str	Δν^Str	I^StrC	I^StrC/I^Strm
a Values of ν^Str and I^Str are given in cm^−1 and km mol^−1 respectively.
(H^+δ–C)-(23)-A	3422.0	−13.2	84.70	1.2
(H^+δ–C)-(23)-B	3422.0	−13.2	84.70	1.2
(Be–H^−δ)-(23)	2041.8	−8.5	43.9	43.9
(H^+δ–N)-(24)-A	3727.1	−37.4	312.3	1.2
(H^+δ–N)-(24)-B	3727.1	−37.4	312.3	1.2
(Be–H^−δ)-(24)	2033.4	−16.9	90.0	90.0
(H^+δ–C)-(25)-A	3405.4	−29.8	18.2	3.7
(H^+δ–C)-(25)-B	3405.4	−29.8	18.2	3.7
(Be–H^−δ)-(25)	2073.9	+23.6	0.0	0.0
(H^+δ–N)-(26)-A	3689.2	−75.3	1.4	1.4
(H^+δ–N)-(26)-B	3689.2	−75.3	1.4	1.4
(Be–H^−δ)-(26)	2096.1	+45.8	0.0	0.0

---

Through the application of the B3LYP/6-311++G(3df,3pd) calculations, Fig. 14 presents the optimized geometries of the BeH₂⋯HCF₃ (27), MgH₂⋯HCF₃ (28), LiH⋯HCF₃ (29), and NaH⋯HCF₃ (30) dihydrogen-bonded complexes. The deformations in the C–H^+δ bonds of the fluoroform can be clearly seen by means of the results of −0.001, 0.0002, 0.003, and 0.004 Å related to the 27, 28, 29, and 30 dihydrogen-bonded complexes respectively. Although the bond lengths on the hydrides are altered due the complexation, we focused here on the structural changes occurring in the C–H^+δ bonds. A shortening of −0.001 Å in C–H^+δ of the fluoroform was thus computed only in 27. In contrast to this, slight elongations in the bond lengths of C–H^+δ with variations between 0.0002–0.004 Å were obtained after the formation of 28, 29, and 30. It can be seen that reduction trends in C–H^+δ have been observed when longer R_{(H^−δ⋯H^+δ)} dihydrogen bonds occur, specifically with the value of 2.215 Å in 27. On the other hand, enlargements in the bond lengths of C–H^+δ were observed in the 28, 29, and 30 systems, which are stabilized by the shortest dihydrogen bond distances.⁶⁴²

Fig. 14 Optimized geometries of the BeH₂⋯HCF₃ (27), MgH₂⋯H CF₃ (28), LiH⋯HCF₃ (29), and NaH⋯HCF₃ (30) dihydrogen-bonded complexes using B3LYP/6-311++G(3df,3pd) calculations. Values of the bond length enhancements are given in parentheses. The values of the C–H, Mg–H, Be–H, Li–H, and Na–H bond lengths of the monomers (HCF₃, MgH₂, LiH and NaH) are 1.089 Å, 1.700 Å, 1.325 Å, 1.589 Å and 1.876 Å, respectively.

In Table 8 are listed the results of the stretch frequencies and absorption intensities computed for the dihydrogen-bonded complexes 27, 28, 29, and 30 with all calculations performed at the B3LYP/6-311++G(3df,3pd) level of theory. As is well-known, the new infrared vibrational modes emerge by means of the formation of hydrogen-bonded complexes,⁶⁴³ and in addition, these vibrational modes are weak stretch bands and they are often interpreted as being the frequencies of the intermolecular interaction.⁶⁴⁴ As such, but taking into account the revisited dihydrogen complexes in this current work, it should be assumed that these kinds of vibrational modes are related to the dihydrogen bond frequencies, . In this regard, it can be reported that, according to the values of the dihydrogen bond energies (2.31–19.66 kJ mol⁻¹),³⁸⁷ in general, the results obtained at the B3LYP/6-311++G(3df,3pd) level of theory do not corroborate these values for intermolecular energy. In other words, the weakly bonded dihydrogen complex 27, which has the lowest interaction energy ΔE^C(BSSE + ΔZPE) of 2.31 kJ mol⁻¹, also presents a low dihydrogen bond frequency , though not the weakest. On the contrary, the stretch frequency of the dihydrogen bond in the complex 28 is the weakest, but its intermolecular energy ΔE^C(BSSE + ΔZPE) of 6.67 kJ mol⁻¹ is of medium-strength, although higher than the 2.31 kJ mol⁻¹ found in 27. It is greatly important to emphasize that 29 furnishes a stronger stretch frequency of the dihydrogen bond than that observed in 30, whose values are 148.2 and 91.7 cm⁻¹ respectively. On the other hand, their energies ΔE^C of 19.43 and 19.66 kJ mol⁻¹ are equivalent.

Table 8 Values of the stretch frequencies of the BeH₂⋯HCF₃ (27), MgH₂⋯HCF₃ (28), LiH⋯HCF₃ (29), and NaH⋯HCF₃ (30) dihydrogen complexes obtained by using the B3LYP/6-311++G(3df,3pd) calculations^a

Parameters	Dihydrogen complexes
	(27)	(28)	(29)	(30)
a Values of ν^Str and I^Str are given in cm^−1 and km mol^−1 respectively. Values of ν^Str and I^Str (HCF3 monomer) in parentheses.
	73.2	66.0	148.2	91.71
	0.30	0.22	0.33	0.02
	3137.9 (3121)	3116.7 (3121)	3062.3 (3121)	3039.0 (3121)
	3.99 (26.9)	0.57 (26.9)	45.8 (26.9)	57.5 (26.9)
	+16.3	−4.9	−59.3	−82.6
	0.15	0.02	1.70	2.14

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Nevertheless, if the changes in the C–H^+δ bonds of the fluoroform are taken into account, there is an interesting observation with regard to the 27, 28, 29, and 30 dihydrogen-bonded complexes. As debated above, no satisfactory correlation between dihydrogen bond energy and vibrational modes has been obtained, but in terms of modifications in C–H^+δ, distinct vibrational shifts have been verified on this bond. In accordance with recent studies published in the specialized literature,³⁸⁷ it was extremely unexpected to observe the red-shifts of −4.9, −59.3, and −82.6 cm⁻¹ in the 28, 29, and 30 dihydrogen-bonded complexes, of which the last two showed an increase in absorption intensity, whose ratio values are 1.70 and 2.14 respectively. Meanwhile, a blue-shift of +16.3 cm⁻¹ was detected in 27, although a reduction in absorption intensity of 0.15 has been also observed. In terms of intermolecular strength, it can be confirmed that the higher values of dihydrogen bond energy in 28, 29, and 30 are associated with larger red-shifts, whereas the weak intermolecular contacts are not related either to longer or shorter red-shifts. In fact, it is evident that the dihydrogen-bonded complex 27, in addition to complexes 28, 29, and 30, they coexist under the formation of red- and blue-shifted dihydrogen bonds³⁸⁷ respectively.

Regarding the ternary dihydrogen complexes (32 and 33), their dihydrogen bond distances are longer in comparison with the binary complexes. In a contrasting reflection on this but already discussed here, it is well-known that ternary hydrogen complexes formed by π or lone-electron pairs as proton acceptors are much more stable than binary complexes in terms of their intermolecular distances.^{213,378,397,398,639,645,646} Unfortunately, this observation is not verified here, but it is common because increases in double or bifurcate dihydrogen bond distances are well-known,^555,556,647e.g., dihydrogen-bonded systems formed by ethyl cation and beryllium hydride.³⁹⁸ Nevertheless, it can be seen that the values of 2.1404 Å (32) and 1.7934 Å (33) for H^−δ⋯H^+δ are shorter than the sum of the van der Waals radii for the hydrogen atom, whose tabulated datum is 1.09 Å. Moreover, a novel interaction, the so-called alkali–halogen bond (the alkali being sodium), occurs between sodium and fluorine (Na⋯F) or chlorine (Na⋯Cl). Once again, this statement can be demonstrated by the van der Waals radii of Na (2.27 Å), F (1.47 Å), and Cl (1.75 Å), whose summed values of 3.74 Å (Na + F) and 4.02 Å (Na + Cl) are longer than the corresponding results of 2.4345 Å (32) and 2.9000 Å (33) shown in Fig. 15. In line with these findings, significant enhancements of 0.0386 Å and 0.0461 Å in the C–W bonds (W = F or Cl) were noted only in the ternary systems. On comparing binary with ternary complexes, however, a change in the sodium hydride could be evidenced. The Na–H bond shows a reduction in its length due to the formation of the binary complexes NaH⋯HCF₃ (30) and NaH⋯HCCl₃(31). Otherwise, an increase in the Na–H bond length was detected in the ternary systems 32 and 33.

Fig. 15 Optimized geometries of the ternary complexes NaH⋯2(HCF₃) (32) and NaH⋯2(HCCl₃) (33) obtained from B3LYP/6-311++G(3df,3pd) calculations.

These contrasting structural profiles certainly lead to very distinct vibrational effects, which if hydrogen bonds, dihydrogen bonds,^{190,648–651} or even the alkali–halogen bonds are independently formed, the characterization of the chemical shifts either from red or blue origin is the spectroscopy cornerstone for identifying the formation of intermolecular interactions. There is a trend toward weakness in the bonds of the proton donors (H–X) of HCCl₃ from NaH⋯HCCl₃ up to the formation of 33. Thus, the stretch frequency of the oscillator ν_C–H is downward shifted by −81.00 cm⁻¹ (δν_C–H,33 − δν_C–H,31) followed by an absorption intensity ratio of 2.4, whose values are also given in Table 9. On the other hand, HCF₃ behaves as a blue-shifted proton donor because the value of its stretch frequency ν_C–H is upward shifted by +42 cm⁻¹ (δν_C–H,32 − δν_C–H,30), which is a spectroscopic characteristic widely-established for fluoroform.^376,377 In total corroboration with recent investigations,⁶⁵² it is demonstrated that intermolecular complexes with shorter distances such as those observed in 33 are identified by red-shift events on the stretch frequencies of their proton donors. In contrast to this, stretch frequencies of longer intermolecular distances are shifted to blue, e.g., complex 31.

Table 9 Structural and vibrational parameters of the NaH⋯HCF₃ (30) and NaH⋯HCCl₃ (31) binary dihydrogen complexes, as well as for of the NaH⋯2(HCF₃) (32) and NaH⋯2(HCCl₃) (33) ternary alkali–halogen complexes obtained through the B3LYP/6-311++G(3dp,3df) calculations^a

Parameters	Hydrogen complexes
	(30)	(31)	(32)	(33)
a Values of ν and I are given in cm^−1 and km mol^−1, respectively. Values of R and r are given in Å. X = CF3 or CCl3; W = F or Cl. Values of ν(Na–H^−δ) and I(Na–H^−δ) for the sodium hydride monomer are 1167.93 cm^−1 and 215.93 km mol^−1, respectively. Values of ν(H–X) and I(H–X) for the fluoroform monomer are 3121.66 cm^−1 and 26.95 km mol^−1, respectively. Values of ν(H–X) and I(H–X) for the chloroform monomer are 3187.48 cm^−1 and 1.67 km mol^−1, respectively.
δr(Na–H^−δ)	−0.0006	−0.0020	0.0785	0.0925
δr(H–X)	0.0046	0.0166	0.0027	0.0218
δr(C–W)	0.0063	0.0048	0.0386	0.0461
R(H⋯H)	1.9090	1.7250	2.1404	1.7934
R(Na⋯W)	—	—	2.4345	2.9000
ν(Na–H^−δ)	1245.80	1265.62	1079.76	1031.31
I(Na–H^−δ)	351.7	539.35	565.80	233.62
δν(Na–H^−δ)	+77.87	+97.69	−88.17	−136.62
I(Na–H^−δ),c/I(Na–H^−δ),m	1.63	2.49	2.62	1.08
ν(H–X)	3039.01	2929.71	3080.95	2848.7
I(H–X)	57.52	510.93	52.47	1228.03
δν(H–X)	−82.65	−257.77	−40.71	−338.77
I(H–X),c/I(H–X),m	2.13	305.94	1.94	735.34
ν(C–W)	686.51	366.74	683.46	361.29
I(C–W)	20.91	0.68	43.60	0.61
δν(C–W)	−6.69	+1.03	−9.74	−3.42
I(C–W),c/I(C–W),m	1.70	4.53	3.55	4.06
ν(H^−δ⋯H^+δ)	91.71	97.68	69.16	81.28
I(H^−δ⋯H^+δ)	0.02	0.12	0.34	0.00(1)
ν(Na⋯W)	—	—	47.54	35.89
I(Na⋯W)	—	—	1.06	0.65

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Besides the chemical shifts, Table 9 also lists the values for the new vibrational modes, commonly referred as intermolecular infrared stretch frequencies and their absorption intensities for the systems 30, 31, 32 and 33. Applying the same comparative procedure of analysis used for the binary and ternary systems, blue-shifts in sodium hydride were determined, whose values for 30 and 31 were +77.87 cm⁻¹ and +97.69 cm⁻¹, respectively. In contrast, red-shifts were identified in the H–X oscillator for which, although the value of −257.7 cm⁻¹ in 31 is large, the result of −82.65 cm⁻¹ for 30 is unusual because fluoroform is a standard proton donor model with blue-shift characteristic.³⁸¹ Hence, some chemical shifts, such as the red of −6.69 cm⁻¹ and the blue of +1.03 cm⁻¹, are inconsistent with the observation of the systematic increase in the C–F and C–Cl bond lengths of 30 and 31. Although the C–F and C–Cl bond lengths are those with the largest variations in the ternary complexes, their stretch frequencies are slightly downward shifted, corresponding to lower red-shift values of −9.74 and −3.42 cm⁻¹ for 32 and 33, respectively. These red shifts are associated with those observed for Na–H, in which the computed values of −88.17 and −136.62 cm⁻¹ for 32 and 33 are surprising because blue-shift effects of +77.87 and +97.69 cm⁻¹ have been previously observed for this oscillator upon the formation of the binary systems 30 and 31. All of these red- and blue-shift effects can be explained through structural analysis, i.e., elongation and reduction of the corresponding bond length. For both sodium hydride and halocarbon bonds (C–F or C–Cl) it should be noted that no correlation between the magnitude of the red- and blue-shifts and the absorption intensity ratios was found. Besides the downward shift in the stretch frequencies, the specialized literature reports that red-shift effects for proton donors are often characterized by an increase in the absorption intensities, which in this study may also occur in the case of blue-shift effects. In other words, there is no tendency for chemical shifts, of red or blue origin, according to the absorption intensity.

In relation to the stretch frequencies of the dihydrogen bonds, their values are in good agreement with the intermolecular distances. Note that R_{(H^−δ⋯H^+δ)} is shorter in 31, although its value of 97.68 cm⁻¹ is slightly higher than the 91.71 cm⁻¹ of 30. In satisfactory agreement with traditional bound complexes,⁶³⁰ the stretch frequencies presented extremely low absorption intensities, which are spectroscopically inherent to any intermolecular interaction type. In conformity with the results of the intermolecular distances debated elsewhere, the ternary complexes presented weak dihydrogen bond stretch frequencies, whose values of 69.16 cm⁻¹ (32) and 81.28 cm⁻¹ (33) are accompanied by the lowest results for absorption intensity. With regard to the stretch frequencies of the alkali-halogen bonds, their values calculated by the B3LYP/6-311++G(3df,3pd) level associated with other levels show a direct correlation with the bond lengths,⁶⁵³ as can be seen by the linear coefficient R² of 0.95 graphically illustrated in Fig. 16.

Since ν^Str_(Na⋯F) and ν^Str_(Na⋯Cl) are modes containing low absorption intensities in the range of 0.65–1.06 km mol⁻¹, their localizations on the vibrational spectrum could only be obtained following preliminary structural identifications. Thus, although both alkali–halogen bonds Na⋯F and Na⋯Cl have been discovered, they are not sufficiently strongly bound to overestimate the stabilization of the ternary complexes. Therefore, the H^−δ⋯H^+δ dihydrogen bonds are still predominant in this regard.⁶⁵²

Fig. 16 Relationship between the values of the new vibrational modes and intermolecular distances obtained from B3LYP/6-311++G(3df,3pd) calculations.

The intermolecular QTAIM topology of the π⋯H hydrogen bonds

The bond paths (34–34′) presented in Fig. 17 show all the BCPs for the hydrogen complexes C₂H₂⋯HF and C₂H₂⋯2(HF). As mentioned previously, these BCPs are located by means of topological calculations,^654–658 by which are computed the electron density (ρ) and its Laplacian field (∇²ρ),⁶⁵⁹ whose values are listed in Table 10. According to the ∇²ρ values, the H–F bonds are fully characterized as shared interactions in the 2(HF) dimer, as well as in the C₂H₂⋯HF and C₂H₂⋯2(HF) complexes. With regard to the hydrogen bonds, these interactions are defined as closed shell because the positive results of ∇²ρ are in the range of 0.026 up to 0.119 e a_o⁻⁵. In terms of electronic density, the values obtained for all hydrogen bonds are very low. For instance, in the C₂H₂⋯2(HF) hydrogen complex, the respective values of electronic density for un-π⋯H and F^d⋯H^e are 0.023 and 0.029 e a_o⁻³. However, the π⋯H interaction in the C₂H₂⋯HF hydrogen complex presents a lower electronic density of 0.018 e a_o⁻³. This indicates that charge concentrations are located on the chemical bonds, whereas electronic depletion is related with the π⋯H interaction,⁶⁶⁰ as can be seen from the contour relief map (35) of C₂H₂⋯2(HF) shown in Fig. 18.

Fig. 17 QTAIM bond paths for the C₂H₂⋯HF (34) and the C₂H₂⋯2(HF) (34′) hydrogen-bonded complexes.

Table 10 Values of electronic densities ρ and Laplacians ∇²ρ for all interactions in the 2(HF), C₂H₂⋯HF (34), C₂H₂⋯2(HF) (34′), C₂H₄⋯HF (36), and C₂H₄⋯2(HF) (36′) hydrogen complexes obtained by using QTAIM calculations^a

Parameters	Hydrogen complexes
	2(HF)	(34)	(34′)	(36)	(36′)
a Values of ρ and ∇^2ρ are defined as e ao^−3 and e ao^−5 respectively.
ρ(H^c–F^d)	0.362	0.354	0.339	0.353	0.336
ρ(H^e–F^f)	0.358	—	0.349	—	0.345
ρ(C^1=C^2)	—	—	—	0.342	0.343
ρ(C^1≡C^2)	—	0.411	0.410	—	—
ρ(π⋯H)	—	0.018	—	0.017	0.024
ρ(un-π⋯H)	—	—	0.023	—	—
ρ(p-π⋯H)	—	—	—	—	—
ρ(F^d⋯H^e)	0.024	—	0.029	—	0.040
ρ(F^f⋯H^b)	—	—	0.007	—	0.021
∇^2ρ(H^c–F^d)	−2.776	−2.65	−2.51	−2.63	−2.30
∇^2ρ(H^e–F^f)	−2.722	—	−2.64	—	−2.43
∇^2ρ(C^1=C^2)	—	—	—	−1.02	−1.00
∇^2ρ(C^1≡C^2)	—	−1.24	−1.23	—	—
∇^2ρ(π⋯H)	—	0.053	—	0.050	0.019
∇^2ρ(un-π⋯H)	—	—	0.061	—	—
∇^2ρ(p-π⋯H)	—	—	—	—	—
∇^2ρ(F^d⋯H^e)	0.107	—	0.119	—	0.137
∇^2ρ(F^f⋯H^b)	—	—	0.026	—	0.141

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Fig. 18 Relief map of the C₂H₂⋯2(HF) (35) hydrogen-bonded complex.

The structural results and the infrared analysis for the C₂H₂⋯2(HF) complex suggest that a new hydrogen bond may be formed, R_(F^f⋯H^b) and ν_(F^f⋯H^b), but this interaction has not been confirmed yet. Nevertheless, the QTAIM calculations have categorically identified the F^f⋯H^b interaction, as can be seen by the BCP_b–f (34′) illustrated in Fig. 17. Comparing the electron density of π⋯H, F^d⋯H^e, and F^f⋯H^b, the lowest value of 0.007 e a_o⁻³ for F^f⋯H^b defines it as a closed shell interaction, as stated by Popelier, Bader, Matta and others.^661–665 Moreover, the typical condition of the Laplacian field guarantees the formation of F^f⋯H^b, i.e., ∇²ρ > 0. For the formation of the complex C₂H₂⋯2(HF), the QTAIM calculations provide essential information regarding the π⋯H interaction, which thus requires special attention. In the discussion of the structure of the C₂H₂⋯HF hydrogen complex, it was noted that the π⋯H interaction is formed exactly at the midpoint of the C¹≡C² bond of the acetylene and QTAIM calculations, locating the bond path⁶⁶⁶ and BCP_1–2 (34), as illustrated in Fig. 17, corroborates this.⁶⁰³ On the other hand, the QTAIM calculations for the C₂H₂⋯2(HF) hydrogen complex show that π⋯H seems not to be well defined in the T-shaped form. Surely, the uncommon un-π⋯H interaction, hitherto represented as H^c⋯C², is characterized by an electronic density of 0.023 e a_o⁻³ and a Laplacian field of 0.061 e a_o⁻⁵, although it is formed by the interaction between the H^c hydrogen and the C² atoms, as can seen in the BCP_c–2 of Fig. 17 (34′). Some of these results have also been reported by Grabowski and Ugalde⁶⁶⁷ in an investigation in which it is explained whether carbon atoms or BCP can be proton acceptors. In confrontation with this finding, Tsuzuki et al.⁶⁶⁸ also have performed theoretical calculations at MP2 level as well as more sophisticated ones such as CCSD(T) in works elaborated to investigate the nature of these proton acceptor types. Since then the possibility of π charge transference to the σ* anti-bonding orbital of the HF molecule in the C₂H₂⋯2(HF) complex has been practically ruled out, instead there is a prominent redistribution of the electron density throughout the ‘intramolecular’ C₂H₂⋯2(HF) complex under the following conditions: σ → σ* (C²–H^c), σ → σ* (F^d–H^e), and σ → σ* (F^f–H^b).⁶⁰³

Fig. 19 shows the bond paths and BCP for the C₂H₄⋯HF (36) and C₂H₄⋯2(HF) (36′) hydrogen-bonded complexes. Once again, the results organized in Table 10 reveal that the electronic densities on the C=C bonds are slightly changed, i.e., values of 0.342 and 0.343 e a_o⁻³ for the C₂H₄⋯HF binary and C₂H₄⋯2(HF) ternary complex, respectively. On the other hand, there is evidence of a significant variation in the electronic density on the hydrofluoric acid, whose values are 0.353 and 0.336 e a_o⁻³ for H^c–F^d from C₂H₄⋯HF to C₂H₄⋯2(HF), and 0.345 e a_o⁻³ for H^e–F^f in the ternary complex. In terms of intermolecular interactions, the electronic density values are extremely low, which vary between 0.017 and 0.024 e a_o⁻³. In association with the positive values of the Laplacian, these electronic densities indicate that a hydrogen bond is formed within each intermolecular BCP,⁶⁶⁹ as can be seen from the BCP_c–π in 36 and 36′ shown in Fig. 19. Although, as mentioned previously, a ternary interaction between the second hydrofluoric acid (H^e–F^f) and hydrogen atoms (H^a and H^b) of ethylene may be structurally feasible, the identification of the BCP_b–f and BCP_a–f (see Fig. 19 once again) and therein the calculation of the unique Laplacian value of 0.141 e a_o⁻³ characterizes both of the interactions F^f⋯H^a and F^f⋯H^b as hydrogen bonds. By taking into account the framework of the QTAIM atomic basis,⁶⁷⁰ the hydrogen bonds possess charge density concentrated in separate nuclei (donor and acceptor of protons),⁶⁷¹ as can be seen in the relief map (see Fig. 20) of the C₂H₄⋯2(HF) (37) ternary complex.

Fig. 19 Bond paths for the C₂H₄⋯HF (36) and C₂H₄⋯2(HF) (36′) hydrogen bond complexes.

Fig. 20 Relief map for the C₂H₄⋯2(HF) (37) hydrogen-bonded complex.

The acquisition of the QTAIM protocol and its application is currently widespread in many scientific researches around the world.^672–675 Precisely, the ability of the QTAIM protocol to model the chemical bonds via quantum mechanical arguments^676,677 distinguish it as one of the most efficient theoretical procedures for studying intermolecular systems, such as hydrogen bond complexes and their spectral phenomena of shifted frequencies.^678,679 In regards to the C₂H₄⋯HCF₃, C₂H₃(CH₃)⋯HCF₃, and C₂H₂(CH₃)₂⋯HCF₃ complexes, Table 11 gathers the results for their main QTAIM topological parameters: the electron density ρ and its Laplacian ∇²ρ. Very briefly, the low values of the electronic densities in the range of 0.006–0.007 e a_o⁻³ and the positive Laplacian results explain the formation of hydrogen bonds. As for the hydrogen bond strengths whose structural parameters were previously discussed,^680,681 the highest intermolecular electronic density was found in 40, whereas the lowest were computed for 38 and 39. In the latter case, the electronic density is related to the C²⋯H^c improper hydrogen bond, in which the carbon acts as a proton acceptor, a phenomenon that is not usually observed. Fig. 21 plots the BCPs and bond pathways of the electronic density of complexes 38, 39 and 40. It can be seen that π⋯H^c hydrogen bonds are formed in 38 and 40 through the characterization of the BCP_c–π, whereas an improper C²⋯H^c interaction is typical of 39, when the BCP_c–2 has been located. The chemical bond analyses with high charge concentration whose Laplacian results are negative or which, according to the QTAIM, are shared interactions show that the reduction in electronic density in the C^a=C^b bond is one of the most intense charge transference effects upon the formation of the C²⋯H^c and π⋯H^c hydrogen bonds. The charge transference is understood as one of the electronic cornerstones for the formation of hydrogen bonds,⁶⁸² and this can be demonstrated by the accumulation of electronic density on the C–H^c bonds of fluoroform (see Table 11), where, in contrast to the monomer state, an increase, of 0.303(3), 0.303(1), and 0.303(2) e a_o⁻³, respectively, can be observed in (38), (39), and (40).

Table 11 Values of the electronic densities ρ and Laplacians ∇²ρ for the main bonds of the C₂H₄⋯HCF₃ (38), C₂H₃(CH₃)⋯HCF₃ (39), and C₂H₂(CH₃)₂⋯HCF₃ (40) complexes obtained by using the QTAIM calculations^a

Parameters	Hydrogen complexes
	(38)	(39)	(40)
a Values of ρ and ∇^2ρ are given in e ao^−3 and e ao^−5 respectively. Values of ρ and ∇^2ρ for the C^a=C^b bond of the ethylene, propylene, and t-butylene monomers are given in parentheses. Values of ρ and ∇^2ρ for the H–C bond of the HCF3 monomer are 0.302 e ao^−3 and −1.112 e ao^−5 respectively.
ρ(H^c⋯π)	0.006(0)	—	0.007(0)
∇^2ρ(H^c⋯π)	0.015(5)	—	0.015(7)
ρ(H^c⋯C^2)	—	0.006(9)	—
∇^2ρ(H^c⋯C^2)	—	0.015(4)	—
ρ(C^a=C^b)	0.3430 (0.3437)	0.3417 (0.3428)	0.3410 (0.3419)
∇^2ρ(C^a=C^b)	−1.025 (−1.028)	−1.0110 (−1.017)	−1.0921 (−1.004)
ρ(C–H^c)	0.303(3)	0.303(1)	0.303(2)
∇^2ρ(C–H^c)	−1.138	− 1.139	−1.139

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Fig. 21 BCP and bond paths of the C₂H₄⋯HCF₃ (38), C₂H₃(CH₃)⋯HCF₃ (39), and C₂H₂(CH₃)₂⋯HCF₃ (40) complexes.

The intermolecular QTAIM topology of the H^−δ⋯H^+δ dihydrogen bonds

By consulting the literature, the application of QTAIM to the studies of intermolecular interactions via hydrogen has yielded very interesting results,^683–688 mainly about the interpretation of vibrational modes in halogen–hydride interactions as per a very recent work signed by Mohajeri et al.⁶⁸⁹ For the dihydrogen-bonded complexes re-examined here, being binary or ternary, the topological parameters should be ideally suitable for interpretation of their spectral shifts. Thus, Fig. 22 presents the bidimensional graphs of the Laplacian attributed to the electronic densities of the BeH₂⋯HCl binary complex 41, as well as of all isomeric series 42, 43, and 44 of the BeH₂⋯2(HCl) ternary complex. It can be seen that ∇²ρ > 0 for all intermolecular interactions, while ∇²ρ < 0 for H–Cl bonds. Thus, not only in terms either of the Laplacian or electronic density ρ, the QTAIM topology can also be discussed using through the atomic charges q_(Ω), whose values are summarized in Table 12. In relation to 42, this complex is stabilized by _{(H^−δ⋯H^+δ)}-(42)-A and _{(Cl^−δ⋯H^+δ)}-(42)-B, by which the existence of a dihydrogen–hydrogen system type is unveiled. The values of the intermolecular electronic densities vary from 0.001 to 0.018 e a_o⁻³, and in fact this is a wide relative variation, especially when the second contact is taken into account, because the R_{(Cl^−δ⋯H^+δ)} distance of 3.262 Å is longer than the sum of the van der Waals radii,^{597,601,602,690–693} which for chlorine and hydrogen sums to 3.0 Å.

Fig. 22 Illustration of the Laplacians of the BeH₂⋯HCl binary dihydrogen complex (41), as well as of the ternaries (42), (43), and (44) dihydrogen isomers.

Table 12 QTAIM topological parameters for the binary (41) and ternary isomers (42), (43), and (44) of the BeH₂⋯2(HCl) dihydrogen complex^a

BCP	QTAIM topological parameters
	ρ	∇^2ρ	ΔQ(Ω),(H–Cl)
a Values of ρ and ∇^2ρ for the hydrochloric acid (monomer) are 0.252 e ao^−3 and −0.775 e ao^−5 respectively. Values of ΔQ(Ω) are given in atomic units (a.u.).
(H–Cl)-(41)	0.246	−0.761	−0.030
(H–Cl)-(42)-A	0.245	−0.756	−0.034
(H–Cl)-(42)-B	0.252	−0.777	−0.002
(H–Be)-(42)	0.103	0.122	—
(H–Be)-(42)	0.100	0.127	—
(H–Cl)-(43)-A	0.249	−0.767	−0.023
(H–Cl)-(43)-B	0.249	−0.767
(H–Be)-(43)	0.104	0.123	—
(H–Be)-(43)	0.977	0.128	—
(H–Cl)-(44)-A	0.248	−0.765	−0.026
(H–Cl)-(44)-B	0.248	−0.765	−0.026
(H–Be)-(44)	0.101	0.126	—
(H–Be)-(44)	0.101	0.126	—
(H^−δ⋯H^+δ)-(41)	0.017	0.038	—
(H^−δ⋯H^+δ)-(42)-A	0.018	4.16 × 10^−2	—
(Cl⋯H^+δ)-(42)-B	0.001	4.16 × 10^−2	—
(H^−δ⋯H^+δ)-(43)-A	0.013	0.030	—
(H^−δ⋯H^+δ)-(43)-B	0.013	0.030	—
(H^−δ⋯H^+δ)-(44)-A	0.015	0.035	—
(H^−δ⋯H^+δ)-(44)-B	0.015	0.035	—

---

However, the electronic density of _{(H^−δ⋯H^+δ)}-(42)-A presents a very similar result to _{(H^−δ⋯H^+δ)}-(41), indicating that 42 tends to form a binary complex, rather than a ternary one. With regard to the structure of 43, the formation of double interactions between the two molecules of hydrochloric acid directly aligned on the beryllium hydride demonstrates that H^−δ can be an ‘electronegative hydrogen’.⁶⁹⁴ Although the electronic density value of 0.013 e a_o⁻³ for both the _{(H^−δ⋯H^+δ)}-(43)-A and _{(H^−δ⋯H^+δ)}-(43)-B dihydrogen bonds are characteristic of van der Waals contacts (∼10⁻²–10⁻³ a_o³),^695,696 these interactions are essentially dihydrogen bonds.⁴⁷² Thereby, system 43 is thus formed by two dihydrogen bonds, which provide the formation of three proton donor centres: _{H^−δ(H–Be)}-(43), _{H^+δ(H–Cl)}-(43)-A, and _{H^+δ(H–Cl)}-(43)-B. For this reason, a bifurcate dihydrogen bond is observed. In terms of the structure of 44, two dihydrogen bonds were formed, although in separate hydride elements (H^−δ). Even though _{(H^−δ⋯H^+δ)}-(44)-A and _{(H^−δ⋯H^+δ)}-(44)-B are not the strongest interactions, the electronic density of 0.015 e a_o⁻³ and Laplacian of 0.035 e a_o⁻⁵ indicate that 44 is formed by four proton donor sites: _{H^+δ(H–Cl)}-(44)-A, _{H^+δ(H–Cl)}-(44)-B, _{H^−δ(H–Be)}-(44), and _{H^−δ(H–Be)}-(44).

The remaining proton donors, HCN and HNC, have been also used for the study of the ternary dihydrogen-bonded complexes, in which, the QTAIM calculations were performed for the purpose of topological characterization of the BeH₂⋯2(HCN) (45) and BeH₂⋯2(HNC) (46) bifurcate structures as well as the HCN⋯BeH₂⋯HCN (47) and CNH⋯BeH₂⋯HNC (48) linear ones illustrated in the Fig. 23. Governed by the virial theorem of the QTAIM methodology,^697–699 by which the kinetic (G) and potential (U) energies are embodied into the electronic energy density (H) according to eqn (11), the topological integrations are carried out in order to model a surface map of ρ (see Fig. 24) with identification of the charge density pathways.^{7,502b,666,700–702}

H = G + U (11)

Analysis of the data gathered in Table 13 reveals that no variation has occurred in electronic density when the monomers and the dihydrogen complexes are compared; for example, the values of ρ for the H^+δ–C and H^+δ–N are 0.295 and 0.343 e a_o⁻³, but on the bifurcate 45 and 46 these values are 0.295 and 0.343 e a_o⁻³, as well as 0.294 and 0.341 e a_o⁻³ on the linear 47 and 48, respectively. On the other hand, a reduction of the electronic density of 0.102 e a_o⁻³ on the Be–H^−δ bond was observed, mainly in the BeH₂⋯2(HCN) and BeH₂⋯2(HNC) bifurcate complexes, whose electronic density values are 0.098 and 0.096 e a_o⁻³. Moreover, low quantities of ρ and positive values of ∇²ρ are observed for the whole series of H^−δ⋯H^+δ dihydrogen bonds. Thus, the BeH₂⋯2(HCN), BeH₂⋯2(HNC), HCN⋯BeH₂⋯HCN, and CNH⋯BeH₂⋯HNC complexes can be identified as intermolecular systems because charge density is accumulated in separate nuclei, which are the hydrogen atoms (H^+δ) and the hydride elements (H^−δ).^703,704

Table 13 Values of the QTAIM topological parameters (ρ and ∇²ρ) for the BeH₂⋯2(HX) and XH⋯BeH₂⋯HX (X = CN and NC) ternary dihydrogen complexes^a

BCP	QTAIM topological parameters
	ρ	∇^2ρ
a Values of ρ and ∇^2ρ are given in e ao^−3 and e ao^−5 respectively.
(H^−δ⋯H^+δ)-(45)-A	0.006	0.016
(H^−δ⋯H^+δ)-(45)-B	0.005	0.015
(H^−δ⋯H^+δ)-(46)-A	0.010	0.028
(H^−δ⋯H^+δ)-(46)-B	0.010	0.028
(H^−δ⋯H^+δ)-(47)-A	0.008	0.021
(H^−δ⋯H^+δ)-(47)-B	0.008	0.021
(H^−δ⋯H^+δ)-(48)-A	0.013	0.034
(H^−δ⋯H^+δ)-(48)-B	0.013	0.034
(H^+δ–C)-(45)-A	0.295	−1.176
(H^+δ–C)-(45)-B	0.295	−1.176
(Be–H^−δ)-(45)	0.098	0.125
(H^+δ–N)-(46)-A	0.343	−2.217
(H^+δ–N)-(46)-B	0.343	−2.217
(Be–H^−δ)-(46)	0.096	0.129
(H^+δ–C)-(47)-A	0.294	−1.178
(H^+δ–C)-(47)-B	0.294	−1.178
(Be–H^−δ)-(47)	0.101	0.124
(H^+δ–N)-(48)-A	0.341	−2.223
(H^+δ–N)-(48)-B	0.341	−2.223
(Be–H^−δ)-(48)	0.101	0.128

---

Fig. 23 Illustration of the bond paths of the bifurcate BeH₂⋯2(HCN) (45) and BeH₂⋯2(HNC) (46) as well as the linear HCN⋯BeH₂⋯HCN (47) and CNH⋯BeH₂⋯HNC (48) ternary dihydrogen-bonded complexes.

Fig. 24 Map of the gradient vector of the electron density for the BeH₂⋯2(HCN) ternary dihydrogen-bonded complex.

One of the most expert and influential chemists in hydrogen bonding studies, Leland Cullen Allen,⁷⁰⁵ often has declaimed in its reports that the analysis of intermolecular properties should be performed by accuracy, careful criticism, prudent decision, and prospects.⁷⁰⁶ In order to improve this conjecture, several results point to a close relationship between the structural and topological properties.^707,708 For example, one of the most important investigations, in either intramolecular hydrogen bonds,^709–713 dihydrogen bonds,⁷¹⁴ or π hydrogen bonds,^715,716 is the prediction of intermolecular strength.^{133,300,717–721} Some time ago, vibrational shifts in proton donors were used to rationalize and improve the prediction of hydrogen bond strength.^722,723 By the fact that the QTAIM electronic densities can be used to estimate the variations in the stretch frequencies,⁷²⁴ it is prudent to expound this statement in the analysis of red-shift effects in the systems reviewed here. Fig. 25 shows a graph illustrating the relationship between the red- and blue-shifts and the variations in electronic densities within the BCP of the Be–H^−δ bond of the BeH₂⋯2(HCN), BeH₂⋯2(HNC), HCN⋯BeH₂⋯HCN, and CNH⋯BeH₂⋯HNC complexes. This relationship is validated by eqn (12) presented below:

Fig. 25 Relationship between the harmonic vibrational shifts (red or blue) and variations on the QTAIM electronic densities for the (Be–H^−δ) bonds upon the formation of the ternary dihydrogen-bonded complexes BeH₂⋯2(HCN), BeH₂⋯2(HNC), HCN⋯BeH₂⋯HCN, and CNH⋯BeH₂⋯HNC.

It is perceived that the more accentuated reductions on the electronic density of the proton donors of BeH₂⋯2(HCN) and BeH₂⋯2(HNC) are directly related to their red-shifts, whereas blue-shift events can be explained by slight changes on the charge density. Only for mention, HCN⋯BeH₂⋯HCN and CNH⋯BeH₂⋯HNC are representative complexes for this latter statement. The concentration of electron density is clearly the source of the bond strength, and in this case, small Δρ values explain the blue-shift effect, indicating that the concentration of charge density is not substantially perturbed or, in other words, the strength of the Be–H^−δ bond is preserved. As for the structural parameters, a shortening of the Be–H^−δ bond length occurs due to the formation of BeH₂⋯2(HCN) and BeH₂⋯2(HNC), which is in good agreement with the vibrational analysis and QTAIM topography results already discussed. However, despite the success of electronic density quantification and Laplacian analysis in studies of intermolecular systems, the atomic charge⁷²⁵ is another important QTAIM parameter in this regard, so much that it has been used in the structuring of methods employed to compute the atomic charge distribution, such as the Voronoi Deformation Density (VDD) approach,⁷²⁶ for instance. In contrast to the theoretical formalism,^727–731 the rigorous concept of the QTAIM atomic charge is founded on a purely physical basis,^732–740 namely as the own electronic density:⁷⁴¹

In a paper published in 1970,⁷⁴² Hassel argued for the importance of charge transfer during the formation of interatomic interactions. Some years later, Orville-Thomas et al.^743–746 have observed that the vibrational properties of hydrogen complexes could be predicted by measurement of charge transfer. Therefore, according to the values of the QTAIM atomic charges Q_(Ω) for the series of BeH₂⋯HX (X = F, Cl, CN, and CCH) complexes, the linear coefficient R² of 0.99 obtained in concordance with eqn (14) would appear to describe a good association between the charge transfer and red-shifts of the HX molecules, as can be seen in Fig. 26. In comparison with the observations described by Solimannejad and Alkorta³²¹ regarding the spectrum shifting and charge transference amounts of dihydrogen-bonded complexes, it can also be observed that the high values of Q_(Ω) accord well with large displacements and vice-versa. In spite of this, similar results were obtained for π hydrogen bonded complexes formed by acetylene and monoprotic acids,^273,274 albeit with the application of the Charge–Charge Flux-Overlap (CCFO)^{537,747–750} to compute the corrected charges.⁷⁵¹

Δν^Str_H–X = 7324.4[ΔQ_(Ω), H–X] + 61.1, R² = 0.99 (14)

Fig. 26 Relationship between intermolecular charge transfer (ΔQ_(Ω),H–X) and the red-shift effects (Δν^Str_H–X) of the monoprotic acids for the BeH₂⋯HX dihydrogen-bonded complexes with X = F, Cl, CN, and CCH using B3LYP/6-311++G(3d,3p) calculations.

At this current time, the QTAIM theory is considered one of the most used computational schemes in studies of electronic structure^752–755 because the positive values of the Laplacians of ρ (∇²ρ) accompanied by low quantities of electron density bring the ideal information to characterize the chemical bond, in particular its strength and shape.⁷⁵⁶ By this insight, the results organized in Table 14 indicate that the H^−δ_b⋯H^+δ interaction can be identified as an intermolecular contact of a closed-shell, or, more specifically, a dihydrogen bond. The contour maps and BCPs of 49 and 50 can be seen in Fig. 27, which also illustrates the lines of electronic density and molecular pathway between the interactive atoms, such as for example the H^−δ_b of BeH₂ and MgH₂, as well as the H^+δ of the fluoroform. Since the magnitude of the charge density on the BCP provides a measure of the interaction strength on H^−δ_b⋯H^+δ, it is fair to say that 49 is the weakest bound dihydrogen complex, whereas 50, 51, and 52 are the strongest ones (see relief maps for 51 and 52 illustrated in the Fig. 28). In terms of shared interactions, it can be noted that the electronic charge density on the C–H^+δ bond of the fluoroform is higher in comparison with the value calculated for the H^−δ_b⋯H^+δ dihydrogen bonds. The accumulation of charge density can be explained by the ∇²ρ results, which are less negative for the 49, 50, 51, and 52 complexes than for the fluoroform subunit, in which ∇²ρ is −2.124 e a_o⁻⁵. Moreover, it is worthwhile to emphasize that the value of the charge density on C–H^+δ is higher in comparison with the isolated fluoroform, in which the ρ value is 0.302 e a_o⁻³.

Table 14 Values of the QTAIM topological parameters of the BeH₂⋯HCF₃ (49), MgH₂⋯HCF₃ (50), LiH⋯HCF₃ (51), and NaH⋯HCF₃ (52) dihydrogen complexes^a

Parameters	Dihydrogen complexes
	(49)	(50)	(51)	(52)
a Values of ρ and ∇^2ρ are given in e ao^−3and e ao^−5 respectively. Values of ΔQ(Ω) are given in atomic units (a.u.). Values of ρ and ∇^2ρ of the C–H^+δ bond of the HCF3 monomer are given in parentheses.
ρ(Hb^−δ ⋯H^+δ)	0.007	0.010	0.015	0.016
∇^2ρ(Hb^−δ⋯H^+δ)	0.018	0.023	0.032	0.033
ρ(C–H^+δ)	0.313 (0.302)	0.312 (0.302)	0.311 (0.302)	0.310 (0.302)
Δρ(C–H^+δ)	0.011	0.010	0.009	0.008
∇^2ρ(C–H^+δ)	−1.261 (−2.124)	−1.265 (−2.124)	−1.273 (−2.124)	−1.266 (−2.124)
ΔQ(Ω)	−0.014	−0.025	−0.046	−0.056

---

Fig. 27 Contour lines of the electronic density (thin), molecular pathways (bold) and BCPs (little squares) of the BeH₂⋯HCF₃ (49) and MgH₂⋯HCF₃ (50) dihydrogen-bonded complexes obtained from QTAIM calculations.

Fig. 28 Relief maps of the LiH⋯HCF₃ (51) and NaH⋯HCF₃ (52) dihydrogen-bonded complexes obtained from QTAIM integrations.

In Table 14 are listed the results of the total atomic charge transference ΔQ_(Ω) in the HCF₃ molecule. First, it can be seen that a systematic trend for ΔQ_(Ω) has been found, where the values vary from −0.014 a.u. in 49 to −0.056 a.u. in 50. Although the strength of the H^−δ_b⋯H^+δ dihydrogen bond was initially measured using the attraction energies ΔE^C(BSSE + ΔZPE), the computation of ΔQ_(Ω) suggests that a good correlation between ΔQ_(Ω) and ΔE^C(BSSE + ΔZPE) may be obtained. In other words, the stronger bound the complex is, such as 52 for instance, the higher the quantity of charge density is, that is transferred from the HOMO orbital of NaH to the LUMO of HCF₃.⁷⁵⁷ Although it is by means of the computation of charge transfer that a true description of how the 49, 50, 51 and 52 dihydrogen complexes are strongly or weakly bound, in terms of infrared vibrational modes, Fig. 29 can illustrate a close relationship between the charge transfer ΔQ_(Ω) and the vibrational shifts in the C–H^+δ bond of the fluoroform.⁷⁵⁸ This result is supported by eqn (15):

It can thus be assumed that the blue-shift of C–H^+δ in 49 and red-shifts in 50, 51, and 52 can be explained by lower and higher charge transference amounts, whose values are −0.014, −0.025, −0.046 and −0.056 e.u., respectively. Surely, this accords well with a charge transfer study registered by Li,⁷⁵⁹ whose conclusion affirms that the elongation (red-shift) and contraction (blue-shift) of the bond lengths of the proton donors are caused by high and low quantities of charge transferred, respectively.

Fig. 29 Relationship between variation on stretch frequencies of fluoroform (HCF₃) and ΔQ_(Ω) charge transfers of the BeH₂⋯HCF₃ (49), MgH₂⋯HCF₃ (50), LiH⋯HCF₃ (51), and NaH⋯HCF₃ (52) complexes using B3LYP/6-311++G(3df,3pd) calculations.

Hybrid systems: unsaturated clouds, dihydrogen bonds, and alkali-halogen bonds

The universe of proton acceptors and donors for intermolecular interactions is immense, and even so the discovering of new hydrogen bond types is currently observed in experimental and theoretical researches.^760–767 In a brief comment about vibrational displacements, there are some intermolecular systems presenting peculiar phenomena,⁶⁴¹ such as rehybridization, structural reorganization, and hyperconjugation.^768–772 For the last one, the structure of the ethyl cation (see Fig. 3) is a classical example, in which the theoretical modelling has demonstrated successfully its preferential configuration.⁷⁷³ In comparison with traditional hydrohalic acids,^{203,212,213,326,327} QTAIM calculations revealed that the ethyl cation acts as a typical proton donor upon the formation of the π⋯H⁺ cationic hydrogen bond.⁷⁷⁴ In accordance with Grabowski,⁷⁷⁵ the π-delocalization exerts a great influence upon hydrogen bond complexations, but unsurprisingly the ethyl cation also serves as proton donor in dihydrogen bond interactions.^645,646 With this insight, the Fig. 30 shows the optimized geometries of the C₂H₅⁺⋯2(MgH₂) (53) and 2(C₂H₅⁺)⋯MgH₂ (54) dihydrogen-bonded complexes, in which it can be seen that two MgH₂ molecules attack the non-localized hydrogen (H⁺) of the ethyl cation (C₂H₅⁺). It is clear that the length of 1.699 Å of the Mg–H^−δ bond in MgH₂ is drastically elongated after complexation, e.g., to 1.736 and 1.723 Å in the 53 and 54 complexes respectively. In terms of intermolecular interactions, or more specifically the distances of the R_{(H^−δ⋯H^+δ)} dihydrogen bonds, although the (54) complex presents the shorter R_{(H^−δ⋯H^+δ)} value of 1.807 Å, it is through this result that the most significant deformations of the MgH₂ structure can be observed, as mentioned above in regards to the Mg–H^−δ bond length.

Fig. 30 Optimized geometries of the C₂H₅⁺⋯2(MgH₂) (53) and 2(C₂H₅⁺)⋯MgH₂ (54) dihydrogen-bonded complexes using the BHandHLYP/6-31G(d,p) level of theory.

It should be pointed out that not only the structural analysis, but also the vibrational spectrum is an useful criterion for indicating whether intermolecular systems are formed or not simply by analyzing the red- or blue-shifts in stretch frequencies. Thus, Table 15 lists the main vibrational modes of the 53 and 54 complexes. Initially, the argument that the Mg–H bond lengths are the most important alterations resulting from the formation of the 53 and 54 complexes is, in fact, also demonstrated by the analysis of the stretch frequencies (ν^Str) and absorption intensities (I^Str). By comparing the value of 1638.3 cm⁻¹ for the MgH₂ monomer, a red-shift of −66 cm⁻¹ can be verified after complexation of 53, followed by a drastic increase in the absorption intensity ratio of 998.5. In 54, a red-shift of −80.2 cm⁻¹ is firstly observed due the distorted polarizability on the MgH₂ monomer, which is not active in the infrared spectrum, i.e., the value of the absorption intensity is zero. In view of this, it can be stated that the red-shifts of −66 and −80.2 cm⁻¹ are in agreement with the distance values for the dihydrogen bonds, which reveal that the shorter distance of R_{(H^−δ⋯H^+δ)} provides the appearance of the larger red-shifts, in particular when the modes of the Mg–H^−δ bond in 54 are analyzed. Moreover, the stretch frequencies do not appear to accord with the dihydrogen bond distance strengths, i.e., the more weakly bound complex 53 has the larger of 125.1 cm⁻¹, whereas 71.9 cm⁻¹ is related to the 54 complex.

Table 15 Values of the stretch frequencies (ν^Str) and absorption intensities (I^Str) of the C₂H₅⁺⋯2(MgH₂) (53) and 2(C₂H₅⁺)⋯MgH₂ (54) dihydrogen-bonded complexes by using the BHandHLYP/6-31G(d,p) calculations^a

Parameters	Dihydrogen complexes
	(53)	(54)
a The values of ν^Str and I^Str are given in cm^−1 and km mol^−1 respectively. The values of ν^Str and I^Str of the MgH2 monomer are given in parentheses.
	1572.3 (1638.3)	1558.1 (1638.3)
	−66	−80.2
	998.5 (0.0)	0.0(0.0)
	998.5	0.0
	125.1	71.9
	42.6	0.0

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The QTAIM analysis has been currently embodied in studies of carbocation compounds.⁷⁷⁶ By taking into account the structural electronic analysis on the ethyl cation performed by Ren et al.,⁷⁷⁷ the QTAIM topological modelling applied in this situation can be used to interpret the formation of the 53 and 54 dihydrogen-bonded complexes. In this context, the values of the topological descriptors are listed in Table 16, and by analyzing the electronic densities computed by means of the QTAIM approach, the ρ_{(H^−δ⋯H^+δ)} dihydrogen bonds of 53 and 54 were thus identified. However, the values of 0.036 and 0.029 e a_o⁻⁵ of the Laplacian also indicate that these dihydrogen bonds are formed, or in other words, these interactions are seen as a closed-shell model. Taking this finding into account, it should be pointed out that the H^−δ⋯H^+δ dihydrogen bonds were also identified using the ρ results of 0.015 and 0.019 e a_o⁻³. In practice, it can be said that the non-localized hydrogen (H⁺) of the ethyl cation binds with the hydrogen hydride H^−δ of MgH₂, but a very interesting finding is that H^−δ can support a double dihydrogen bond interaction in 54, as can be seen in Fig. 31, which shows the QTAIM contour line maps for the electronic densities.⁷⁷⁸ Besides the properties of the ethyl cation and related systems have been well elucidated respectively by Kubas⁷⁷⁹ and Li et al.⁷⁸⁰ in an overview, it can be said that the ethyl cation is an efficient proton donor for dihydrogen bonds.

Table 16 Values of the QTAIM topological parameters of the C₂H₅⁺⋯2(MgH₂) (53) and 2(C₂H₅⁺)⋯MgH₂ (54) dihydrogen complexes^a

Parameters	Dihydrogen complexes
	(53)	(54)
a Values of ρ and ∇^2ρ are given in e ao^−3 and e ao^−5 respectively.
ρ(H^−δ⋯H^+δ)	0.015	0.019
∇^2ρ(H^−δ⋯H^+δ)	0.036	0.029

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Fig. 31 Visualization of the contour lines of the electronic densities of the 2(C₂H₅⁺)⋯MgH₂ (54) dihydrogen-bonded complex using QTAIM calculations.

In addition, as QTAIM applicability reaches beyond the pure theoretical context, as reported by Abramov,⁴⁹⁵ Matta and Arabi,⁴⁹⁶ and others,⁷⁸¹ once again it should be highlighted that the spectroscopic nature of the harmonic vibrational modes previously discussed herein is one of the greatest goals in this regards. In this scenery, Grabowski³⁵⁴ idealized an interpretation for the chemical shifts (red or blue) of proton donors (X–H with X = CF₃ or CCl₃) based on BCPs localized along their bonds. It was established that variations in the atomic radii of carbon and hydrogen are the reason for the red and blue shift effects as follows: δrX > δrH means red-shift while δrX < δrH means blue-shift. Meanwhile, a correlation with Natural Bond Orbital (NBO)^782,783 analysis is suitable to explain the variation on these atomic radii quoted above. Based on the results listed in Table 17, all red-shift values for the H–X oscillator are explained by the increase in the carbon radius in comparison with hydrogen because the carbon s-character (see Table 18) of 30.52% is not considerably enhanced in comparison with the results of 32.63% (30), 32.94% (31), 32.72% (32), and 33.97% (33). Moreover, the carbon polarization of 56.78% is increased to 60.39% (30), 66.52% (31), 60.63% (32), and 67.40% (33). In other words, higher polarization values are associated with larger atomic radii, and hence are the cornerstone to explaining the red shift events in relation to the H–X bond. The blue-shifts in Na–H occur with δrNa > δrH, which is unusual since the very slight increase in the s-character from 99.18% (NaH monomer) to 99.48% in the 30 and 31 complexes cannot be associated with a significant decrease in polarization (% Na) in ranges of 13.23% (monomer) to 9.95% (30) and 9.23% (31). Therefore, the blue shift effect and length reduction are clearly evident for the Na–H bond.

Table 17 Values of the atomic radii determined from the location of the BCP of the monomers, binary dihydrogen complexes, and ternary alkali–halogen systems^a

Systems	Atomic radii
	rNa(Na–H)	rH(Na–H)	rC(C–W)	rW(C–W)	rC(C–H)	rH(C–H)
a All values are given in Å. The atomic radii variations are listed in parentheses.
NaH	0.8772	0.9969	—	—	—	—
HCF3	—	—	0.4571	0.8816	0.7144	0.3563
HCCl3	—	—	0.7790	0.9965	0.7043	0.3567
30	0.9965 (0.1192)	0.8796 (−0.1172)	0.4586 (0.0014)	0.8865 (0.0048)	0.7360 (0.0216)	0.3581 (0.0017)
31	0.9963 (0.1191)	0.8783 (−0.1185)	0.7751 (−0.0039)	1.0049 (0.0084)	0.7388 (0.0345)	0.3577 (0.0010)
32	1.0188 (0.1408)	0.9343 (−0.0625)	0.8963 (0.0171)	1.7065 (0.0187)	0.7373 (0.0229)	0.3347 (−0.0216)
33	1.0239 (0.1467)	0.9432 (−0.0537)	0.7838 (0.0047)	1.0375 (0.0410)	0.7463 (0.0420)	0.3359 (−0.0208)

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Table 18 Values of the NBO parameters of the 30 and 31 binary dihydrogen complexes, as well as the estimated ones of the 32 and 33 ternary alkali–halogen complexes obtained through the B3LYP/6-311++G(3dp,3df) level of theory^a

Parameters	Complexes
	30	31	32	33
a p and s represent the polarization and s-character of the H–X bond, whose values are given in %.
p(H–X)	60.39	66.52	60.63	67.40
s(H–X)	32.63	32.94	32.72	33.97
	42.34	63.62	29.28	78.49
	—	—	16.88	33.72

---

The values for the topography descriptors computed following the QTAIM procedure for the 30–33 systems are listed in Table 19. In agreement with the positive Laplacian values of 0.0328 and 0.0423 e a_o⁻⁵ for the H⋯H dihydrogen bonds of the 30 and 31 binary systems, the Na–H bonds also present the same closed character, whose ∇²ρ values are 0.1304 and 0.1307 e a_o⁻⁵. There are, thus, two simultaneous noncovalent interactions in 30 and 31, namely H^−δ⋯H^+δ and Na–H. Moreover, this contact leads to depletions in the electronic density, with values of 0.0165–0.0341 e a_o⁻³ which are very low. In recent papers, Grabowski et al.^{527,784–786} predicted the covalent character of several binary and ternary system types by assuming a relationship (−G/U) between the positive values of the kinetic electronic energy density (G) and negative values of the potential electronic energy density (U) computed by the QTAIM routine through the localization of the BCP. Not surprisingly, considering the high dihydrogen bond energy of −23.10 kJ mol⁻¹ and the shortest intermolecular distance of 1.7250 Å, it should be expected that the covalence partially dominates the intermolecular zoo of H^−δ⋯H^+δ in 31 instead of a noncovalent profile. Thus, it is stated that the dihydrogen bond systems formed by chloroform are not only more stable or strongly bound, but their intermolecular frameworks belong to the selective class of partially covalent systems.

Table 19 QTAIM (BCP) topological descriptors of the binary dihydrogen complexes (30 and 31), as well as of the ternary alkali-halogen ones (32 and 33) obtained through the Bader's calculations routine^a

Parameters	Complexes
	30	31	32	33
a Values of ρ and ∇^2ρ are given in e ao^−3 and e ao^−5, respectively. Values of G and U are given in atomic units (a.u.). X = CF3 or CCl3; W = F or Cl.
ρ(Na–H)	0.0341	0.0340	0.0301	0.0290
∇^2ρ(Na–H)	0.1304	0.1307	0.1094	0.1068
G(Na–H)	0.0314	0.0315	0.0263	0.0254
U(Na–H)	−0.0303	−0.0304	−0.0252	−0.0241
–G(Na–H)/U(Na–H)	1.0380	1.0399	1.043	1.0526
ρ(H–X)	0.3104	0.2978	0.3118	0.2943
∇^2ρ(H–X)	−1.2660	−1.1610	−1.2834	−1.1386
G(H–X)	0.0228	0.0307	0.0224	0.0301
U(H–X)	−0.3622	−0.3507	−0.3657	−0.3450
−G(H–X)/U(H–X)	0.0630	0.0863	0.0613	0.0874
ρ(C–W)	0.2817	0.1916	0.2576	0.1744
∇^2ρ(C–W)	−0.4129	−0.2553	−0.3959	−0.1979
G(C–W)	0.3229	0.0658	0.0270	0.0600
U(C–W)	−0.7491	−0.1956	−0.0639	−0.1695
−G(C–W)/U(C–W)	0.4311	0.3364	0.4225	0.3540
ρ(H^−δ⋯H^+δ)	0.0165	0.0240	0.0136	0.0243
∇^2ρ(H^−δ⋯H^+δ)	0.0328	0.0423	0.0244	0.0374
G(H^−δ⋯H^+δ)	0.0078	0.0114	0.0058	0.0104
U(H^−δ⋯H^+δ)	−0.0075	−0.0122	−0.0054	−0.0115
−G(H^−δ⋯H^+δ)/U(H^−δ⋯H^+δ)	1.047	0.9333	1.056	0.9036
ρ(Na⋯W)	—	—	0.0131	0.0101
∇^2ρ(Na⋯W)	—	—	0.0775	0.0452
G(Na⋯W)	—	—	0.0163	0.0092
U(Na⋯W)	—	—	−0.0132	−0.0071
−G(Na⋯W)/U(Na⋯W)	—	—	1.2200	1.2885

---

In the case of the ternary complexes, an extension of the parameters discussed in relation to the binary complexes is carried out, although there are four interactions of which two are dihydrogen bonds and two alkali-halogen bonds. In practice, one analysis is performed for the dihydrogen bonds and other to the alkali-halogen bonds. The H^−δ⋯H^+δ dihydrogen bonds were identified by means of the intermolecular electronic density values ρ_{(H^−δ⋯H^+δ)} of 0.0136 e a_o⁻³ (32) and 0.0243 e a_o⁻³ (33) accompanied by their corresponding positive Laplacians ∇²ρ_{(H^−δ⋯H^+δ)} of 0.0244 e a_o⁻⁵ (32) and 0.0374 e a_o⁻⁵ (33). In structural analysis, the presence of the Na⋯F and Na⋯Cl alkali-halogen bonds has been estimated empirically based on the van der Waals radii.⁷⁸⁷ Herein, these interactions are quantum-mechanically validated through the illustration of Fig 32, as well as by means of the ρ_(Na⋯F) and ρ_(Na⋯Cl) quantities of 0.0131 and 0.0101 e a_o⁻³, as well as the positive ∇²ρ_(Na⋯F) and ∇²ρ_(Na⋯Cl) values of 0.0775 and 0.0452 e a_o⁻⁵, respectively. The Na–H bonds remain noncovalent, as can be seen by the ρ_(Na–H) values of 0.0301 (32) and 0.0290 e a_o⁻³ (33). In addition, the Laplacian fields are positive, which suggests noncovalence in Na–H. Generally, if the −G/U relationship is taken into account, the interactions of dihydrogen bonds or sodium hydride bonds seem to be noncovalent in nature. As an exception to this, the H^−δ⋯H^+δ dihydrogen bond in 33 is partially covalent given the −G_{(H^−δ⋯H^+δ)}/U_{(H^−δ⋯H^+δ)} value of 0.9036. Even if dihydrogen bond pairing is formed in a hydride, its intermolecular partial covalent character is not lost in comparison with the binary system (31). Comparing the −G_{(H^−δ⋯H^+δ)}/U_{(H^−δ⋯H^+δ)} values of 0.9333 (31) and 0.9036 (33), clearly NaH⋯2(HCCl₃) is intermolecularly more covalent than NaH⋯HCCl₃, i.e., the contribution of the potential electronic energy density is greater in the dihydrogen bonds of the ternary complex. If the bound distances are evaluated, it can be affirmed that R_{(H^−δ⋯H^+δ)} of 31 < R_{(H^−δ⋯H^+δ)} of 30. For the alkali–halogen bonds, a similar tendency was observed, which is mainly related to the highly noncovalent character. The −G/U values of 1.2200 (32) and 1.2885 (33) indicate a definitive weakness for Na⋯F and Na⋯Cl. In this regard, Fig. 33 plots a linear relationship between the −G/U results and the intermolecular distances computed at the B3LYP/6-311++G(3df,3pd) level of theory.

−G/U = 0.32[R_{(W⋯Na–H^−δ⋯H^+δ)}] + 0.37, R² = 0.96 (16)

Fig. 32 Bond paths, BCP, RCP, and covalent characterization by means of H = G + U (eqn (11)) for the 33 complex.

Fig. 33 Relationship between the −G/U ratios and intermolecular distances obtained from B3LYP/6-311++G(3df,3pd) calculations.

Although the interactions investigated herein presented very distinct energy values, individual examinations should be performed in order to verify their strengths. Some time ago, a new measure of the interaction strength based on structural data in association with the QTAIM topology was proposed.⁷²² Subsequently, we have suggested that good relationships can also be modelled through spectroscopic parameters instead of structural ones.⁶⁷⁸ An equation to predict the interaction strength is conceived by means of the variation in the bond length or in the chemical shift (red or blue) of the proton donor (HX) combined with ρ and ∇²ρ.

The molecular complex parameter (Δmol) can be fitted by structural (δr) or spectroscopy (δν) data. Besides the QTAIM parameters (δρHX = |ρoHX − ρHX| and δ∇²ρHX = |∇²ρoHX − ∇²ρHX|), it can be noted that Δmol was estimated via vibrational chemical shifts (δν = |νoHX − νHX|).⁷⁸⁸ Up to now, the interaction energies estimated herein can be considered indirect values, but this equation can validate them all. Fig. 34 illustrates the relationship between the values for the dihydrogen bond energies⁷⁸⁹ of 30, 31 and others, such as the dihydrogen-bonded complexes LiH⋯HCF₃ (see Fig. 35), LiH⋯HCCl₃, and NaH⋯HF.^181,216

Fig. 34 Relationship between the values of the estimated intermolecular energies and molecular parameter (Δmol).

Fig. 35 Illustration of the relief map of the electronic density of the LiH⋯HCF₃ dihydrogen complex.

Although the ternary complexes are formed by a symmetrical (cooperative) energy distribution, the simultaneous presence of dihydrogen bonds and alkali–halogen leads to the quantification of their interaction energies through the application of eqn (17) being unapproachable. Thus, the binary systems are more suitable in this regard because their intermolecular energies are directly incorporated into a single interaction strength and, as such, other systems were inserted with the purpose to reinforcing the statement debated herein. As is well-known, the dihydrogen complex NaH⋯HF is strongly bonded and almost covalent, either based on its intermolecular energy of −52.15 kJ mol⁻¹ or even the −G/U ratio of 0.5350. Among the complexes examined in this study, only those formed by chloroform yielded a partially covalent character, and therefore they are much less stabilized than NaH⋯HF. In other words, the covalent character is not systematic in terms of interaction energy, and a careful examination of the QTAIM topological parameters is mandatory in this situation. In summary, it can be reported that the remaining bonds (F–C, Cl–C, and C–H) are covalent or topologically shared. The Laplacian values are negative and vary between −0.3959 and −1.2845 e a_o⁻⁵ in 32, and −1.1386 and −0.1979 e a_o⁻⁵ in 33. In line with this, the −G/U values are significantly lower, thus indicating the pure covalence of these bonds. Finally, for the high and low electronic density centers located in BCPs along the intermolecular surfaces examined in this study, it may be necessary to discuss another topological descriptor, namely the Ring Critical Point (RCP), whose values for the ternary complex 33 are shown in Fig. 32. The algebraic difference between BCP and RCP is related to the Poincaré-Hopf relationship^790–794 in terms of non-zero curvatures of ρ at the critical point, yielding the coordinates (3,−1) and (3,+1). According to the QTAIM framework, BCP is an interatomic pathway containing electronic density without chemical bond characteristics and RCP is defined by an intersection of interatomic surfaces that form a ring. According to Castillo et al.,⁵²⁶ from the lowest G values of 0.0049 a.u. and 0.0043 a.u. it can be easily noted that a slower charge density flux is observed for RCP compared with BCPs. Moreover, the electronic densities of 0.0070 and 0.0067 e a_o⁻³ are smaller and the Laplacian fields tend toward annulment, i.e., ρBCP > ρRCP and ∇²ρBCP > ∇²ρRCP.

The relationship between QTAIM descriptors and quantum chemical calculations

In spite of the growth of the theoretical chemistry in the last decades,⁷⁹⁵ one of the benchmarks in this scientific area is absolutely the development of new and modern computational algorithms.^796–804 Specifically in the context of the chemical bond and intermolecular interactions,^55,805–815 a traditional task is to use a series of basis sets in order to reproduce accurately the available experimental data,^816–817 in general those derived from structural and vibrational analysis.²⁰³ Regarding the QTAIM characterization of intermolecular interactions,^{212,818–829} in particular the hydrogen bonds, it has been demonstrated that Pople's split-valence double-ζ incremented by polarization functions^830,831 with higher angular momentum are not appropriate for basis sets to describe the electronic density computed through the QTAIM protocols.⁸³² Otherwise, valence and diffuse functions fairly do it well. Also, it must be important to emphasize that the variability of the basis sets is outweighed by the high-level quantum chemical calculations using Dunning's correlation-consistent basis sets,⁸³³ such as aug-cc-pVTZ⁸³⁴ that has been considered efficient to reproduce the characteristics of intermolecular interactions.^{810,835–840}

There are some works focusing on the application of several ab initio calculations and DFT functionals in order to improve the description of intermolecular interactions via QTAIM calculations,⁸⁴¹ and currently B3LYP, X3LYP, MP2 and CCSD^528,842,843 are equivalent to reproduce experimental results and predict topological properties, as highlighted by Varadwaj.⁶⁰⁵ Aiming to elucidate the relationship or even an explanation for whether QTAIM is seriously dependent on the theoretical level,⁶⁷² Matta reported the following paper entitled: “How dependent are molecular and atomic properties on the electronic structure method? Comparison of Hartree-Fock, DFT, and MP2 on a biologically relevant set of molecules”.⁸⁴⁴ One of the greatest conclusions of this work is the agreement between HF and MP2, wherein the validation of the HF non-correlative results published as yet becomes reasonable. In fulfilment with this, it has been also documented that the QTAIM results obtained from both HF and DFT (depicted by the B3LYP functional) correlate well with those computed at the MP2 level. Of course even the intermolecular systems formed through van der Waals forces^845–847 are appraised by the development of novel density-functional approaches, which have provided excellent results^848–857 many of them better than MP2.⁶¹⁷ In this assertion, Jabłoński and Palusiak⁸⁵⁸ have investigated the relationship between the basis sets and QTAIM descriptors used for the analysis of covalent bonds, wherein it has been demonstrated that HF and B3LYP levels give very similar QTAIM results with respect to the basis set, whereas 6-31G and DZ-quality yielded unsatisfactory results beyond than those obtained by aug-cc-pVTZ. Furthermore, complete basis sets containing high polarization character such as 6-311++G(2df,2pd) and 6-311++G(3df,3pd) are useful for predicting QTAIM parameters. About the hydrogen bonds formed in the F₃CH⋯NH₃, F₃CH⋯NCH, and FCCH⋯NH₃ complexes, however, another study revealed that QTAIM intermolecular properties computed at the BCP are practically independent from both the method and the basis set.⁸⁵⁹

Concluding remarks

Among several types of intermolecular interactions with their molecular properties clearly established,^860–872 nowadays great progresses in hydrogen bonds studies have been published.^873–881 The model Y⋯HX (where Y and X are elements with higher electronegativity than hydrogen) was evolved,⁸⁸² but regardless of the features of the chemical elements, Y is usually assumed to be a region with high electron density, which may have π bonds instead of typical electronegative elements.⁸⁸³ Although it is widely known that the π⋯H hydrogen bond is formed exactly at the midpoint of the C≡C bond of the C₂H₂⋯HF complex, the application of the QTAIM theory to C₂H₂⋯2(HF) reveals an unusual characteristic of the π⋯H interaction. Instead of forming at the midpoint of the C≡C bond of the acetylene, the π⋯H hydrogen bond in the C₂H₂⋯2(HF) complex is formed by the interaction between the C² (acetylene C≡C) and the H^c (first hydrofluoric acid molecule) atoms. Furthermore, it has been demonstrated that a new hydrogen bond, F^f⋯H^b, arises. Thus, the C₂H₂⋯2(HF) hydrogen complex can be considered an ‘intramolecular system’ formed by three hydrogen bonds, H^c⋯C², F^d⋯H^e, and F^f⋯H^b. With regard to the theoretical study of the molecular properties and infrared spectra of the C₂H₄⋯HCF₃, C₂H₃(CH₃)⋯HCF₃, and C₂H₂(CH₃)₂⋯HCF₃ complexes, B3LYP/6-311++G(d,p) calculations reveal a shortening of the C–H bonds of the fluoroform. Besides the red-shifted hydrogen bonds, it has also been observed that the stretch frequencies of their C–H bonds are shifted to higher values after complexation. The charge transfer mechanism has been confirmed by QTAIM calculations, where the electronic density computation has shown a reduction in charge density on the π bonds accompanied by an increase in the H–C bond of the fluoroform.

In terms of the dihydrogen bond, it has been demonstrated that its strength in the BeH₂⋯HX complexes follows the order: BeH₂⋯HF > BeH₂⋯HCl > BeH₂⋯HCN > BeH₂⋯HCCH. This systematic interaction strength was evaluated by means of the molecular changes observed in the isolated subunits in comparison to those arising from the formation of the BeH₂⋯HX dihydrogen-bonded complexes. Such changes have been described in terms of elongation of the bond lengths and vibrational red-shift effects on the monoprotic acids. A suitable explanation of these structural and vibrational phenomena has been found in the direct relationship between the QTAIM charge transfer and the red-shift effects of the proton donor molecules. The formation of ternary and binary dihydrogen-bonded complexes has been also re-examined. According to analysis of the optimized geometries, examination of the molecular topology by using QTAIM calculations, and the interpretation of the infrared spectrum, the most stable complexes derived from BeH₂⋯2(HCl) are linear and formed by four locations of proton donors.

As for other proton donors, such as X = HCN and HNC, the geometries of the BeH₂⋯2HX and XH⋯BeH₂⋯HX dihydrogen complexes are well defined, with the NCH⋯BeH₂⋯HCN and CNH⋯BeH₂⋯HNC linear complexes exhibiting shorter dihydrogen bond distances, whereas the bifurcate BeH₂⋯2(HCN) and BeH₂⋯2(HNC) exhibit longer ones. More interesting deformations have thus been found in linear complexes, such as red-shift effects in the proton donor bonds H^+δ–C and H^+δ–N, as well as blue-shift in the hydride bonds Be–H^−δ. These vibrational effects have been interpreted using the QTAIM parameters, where the stable electronic density conserved the strength of the Be–H^−δ bonds and therefore caused an upward shift in the vibrational oscillator. Furthermore, the blue- and red-shifts in dihydrogen-bonded complexes formed by BeH₂⋯HCF₃, MgH₂⋯HCF₃, LiH⋯HCF₃, and NaH⋯HCF₃ have also been debated. The analysis of equilibrium geometries at the B3LYP/6-311++G(3df,3pd) level of theory has revealed that the main structural deformation occurring in these systems is related to the C–H^+δ bond of fluoroform, which is considered a proton donor. Likewise, changes in hydride compounds have also been examined, although in the case of the infrared spectrum, significant frequency shifts were only observed on the C–H^+δ bond, which took the form of blue-shifts in BeH₂⋯HCF₃, and red-shifts in the MgH₂⋯HCF₃, LiH⋯HCF₃, and NaH⋯HCF₃ dihydrogen-bonded complexes. Apart from characterizing the H^−δ_b⋯H^+δ dihydrogen bonds, the QTAIM approach has been used to complement analysis of the strength of the C–H^+δ bond by using topological parameters and its relationship with the spectroscopy changes. It has been demonstrated that low and high charge transfer values correlate well with blue- and red-shift dihydrogen-bonded complexes respectively. This suggests that these vibrational effects are correctly interpreted whether the quantity of charge transfer between the HOMO and LUMO orbitals is measured, which, in this case, takes the form of a transfer from the hydride compounds (BeH₂, MgH₂, LiH, and NaH) to the fluoroform proton donor (HCF₃). Regarding the ternary dihydrogen complexes, red-shift effects were verified for the more strongly bound complexes whereas the blue-shifts of weaker ones are caused by increases in the polarization of the carbon atom of the proton donors HCF₃ or HCCl₃. As such, and in agreement with the QTAIM foundations, the H⋯H dihydrogen bonds revealed partially covalent intermolecular profiles with strongly bonded complexes, such as NaH⋯HCCl₃ and NaH⋯2(HCCl₃) and, although being less stable than a pure intermolecular covalent complex like NaH⋯HF, the covalence evaluated by the kinetic and potential electronic energy densities plays a decisive role in the rationalization of the interaction strength of dihydrogen systems.

Finally, the structures of uncommon hydrogen-bonded complexes formed by a non-classical ethyl cation and BeH₂ hydrides have been pointed out. The complexes formed were treated as a hybrid system owing to the existence of the π cloud and hydride bonds, Mg–H^−δ, of the ethyl cation and magnesium hydride, respectively. As mentioned earlier, two dihydrogen complexes were investigated rather than one: C₂H₅⁺⋯2(MgH₂) and 2(C₂H₅⁺)⋯MgH₂. Analysis of the structural parameters and vibrational stretch modes revealed that the dihydrogen distances and the new vibrational modes not correlate. However, QTAIM calculations were applied to investigate the formation of bifurcate and linear dihydrogen bonds in competition with the hyperconjugation effect on the ethyl cation, a phenomenon that is responsible for keeping the structure in a stable condition.

Summary and outlook

In recent paper, Weinhold and Klein⁸⁸⁴ inquire: “What is a hydrogen bond?” Notwithstanding after a half-century of hydrogen bond studies,^885–887 the scientific community elected this interaction as one of the most important issues in chemistry, physics, and biology.^888–911 In specific studies of intermolecular interactions,⁹¹² it is also unanimously agreed that hydrogen bonds are a central topic,⁹¹³ although recently dihydrogen bonds have also received a great deal of attention.⁹¹⁴ The quantum chemical description of the hydrogen bonds⁹¹⁵ has been efficiently performed using computational methods, in particular ab initio, DFT, and semiempirical. However, Blanco et al.^916,917 adapted the quantum mechanics of QTAIM^918–922 and its definition of atoms as open systems⁹²³ to interpret atomic interactions, in which the essence of the hydrogen bonding could be examined in terms of classical parameters, such as Coulomb and exchange potentials,⁹²⁴ as well as electronic correlation.⁹²⁵ In spite of all the quantum methods known, the high level researches on the most diversified systems make QTAIM a particular chapter of the theoretical chemistry, although Bader has stated that some reflections about the reality of the QTAIM analysis should be done.⁹²⁶ Independently, it is by the charge density distribution analysis that QTAIM is extremely useful in the study of the chemical bonds and molecular stability, as well as its application to estimate the interaction strength^{133,300,927–930} or to predict covalent features.^{788,931–934} For these reasons the representative group of π hydrogen complexes, dihydrogen complexes, as well as hybrid π dihydrogen complexes revisited here show that QTAIM can be used to study the vibrational shifts in the proton donor bonds of these systems.^935,936

Acknowledgements

I would like to thanks FAPESB, CAPES, and CNPq Brazilian Funding Agencies. This work is dedicated to the memory of Prof. Richard Frederick William Bader (1931-2012).†

Notes and references

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