What Can Be Learnt from a Location of Bond Paths and from Electron Density Distribution

Abstract

A bond path being a line of maximum electron density linking attractors of two atoms is often applied in various studies as a criterion of the existence of numerous interactions such as for example hydrogen, halogen or pnicogen bond. It covers cases of atom-atom energetically stabilized links, from weak van der Waals interactions, through stronger Lewis acid–Lewis base interactions up to covalent bonds. The location of bond paths also allows interpreting mechanisms of interactions and, in general, of chemical reactions. The Quantum Theory of Atoms in Molecules (QTAIM) results are mainly presented here; however they are supported by other approaches as, for example, the Natural Bond Orbitals (NBO) method or the σ-hole concept. The most important orbital-orbital interactions determined from the NBO method and characterizing different types of interactions are presented. The analysis of the distribution of the electron charge density is also performed here for numerous systems; this is shown that the regions of the concentration and depletion of the electron density coincide with the regions of the negative and positive regions of the electrostatic potential. The role of the analysis of the laplacian of the electron density is shown on the basis of numerous interactions.

1 Introduction

The analysis of the electron charge density is one of the most often applied and standard approaches to describe the system investigated; molecule, ion, greater cluster or even crystal [1–3]. This is why the Quantum Theory of Atoms in Molecules (QTAIM) seems to be a useful tool for such analysis since it concerns directly the electron charge density distribution of the system analyzed [4, 5]. One of ideas of the QTAIM approach is the partitioning of 3D electron charge density space into fragments attributed to atoms (atomic basins); the properties of those fragments are often transferable from one system to another one. In such a way it is possible to consider volumes of atoms or to calculate charges integrated over those volumes. This is important that the mentioned here theoretically calculated volumes and charges may have experimental equivalents since it is possible to perform the crystal structure determination to have the experimental electron charge density distribution in crystal and further apply the QTAIM approach [1–3].

However that is not all; the detailed properties of the electron density, ρ(r), of the system considered may be analyzed [5–7]. The critical points (CPs) denoted by the coordinates r_C, are those where the gradient of the electron density, ∇ρ(r_C), vanishes and they correspond to minima, maxima or saddle points of the electron density. The CP is labeled by giving the duo of values (ω,σ), where ω is the rank of CP while σ is its signature. There are the following critical points; (3,−3)—the local maximum often named as attractor is attributed to the position of atom, (3,−1)—the saddle point which often is called the bond critical point (BCP), (3, +1)—the saddle point which is called the ring critical point (RCP) and (3, +3)—the local minimum, i.e. the cage critical point (CCP). The physical interpretation of critical points mentioned here is very well known and it is discussed in numerous monographs and review articles. This is important to announce here that the positions of (3,−3) critical points (attractors) are attributed to the positions of atoms. However there is an excellent agreement between the positions of non-hydrogen attractors and the corresponding nuclei (at least the differences are much smaller than the experimental or theoretical errors) but there is noticeable disagreement between the positions of hydrogen atom attractors (local maxima of the electron density) and their nuclei [8]. This is later discussed in this chapter.

There is another important term useful to describe the distribution of the electron density—the bond path which links pair of attractors [9, 10]. The two gradient paths which originate at the bond critical point and terminate at each of the two attractors define the bond path [11]. In other words the bond path (BP) is a line of the maximum electron density linking the nuclei (more precisely attractors) of two atoms. The bond critical point is that one at the bond path where the electron density attains the minimum value. There are numerous studies on properties and physical meaning of the bond path. It was pointed out that every bond path is accompanied by a virial path [12]. The latter one is a line linking the same nuclei as those connected by the bond path. The virial path is characterized by the maximally negative potential electron energy density thus it is maximally stabilizing with respect to any other neighboring lines. Bader has pointed out that “one may define a bond path operator as a Dirac observable, making the bond path the measurable expectation value of a quantum mechanical operator” [10]. The bond path denotes two atoms are bonded and it is not equivalent to the chemical bond [10]. The former term can describe the complete range of bonded interactions and is attributed to the electron charge accumulation between the pair of atoms, on the other hand the latter term—bond is limited and dominated by the pair-electron concept of Lewis. This is why the term “bond path” at least should be considered as an alternative way to describe the arrangement of atoms in the species analyzed. This is discussed in the next section.

The molecular graph is defined as a set of bond paths and critical points [6, 7]. Figure 15.1 shows the molecular graph of the complex of water with fluorobenzene. One can observe here the bond paths indicating covalent bonds (solid lines) of the fluorobenzene and water molecules as well as two bond paths indicating intermolecular interactions (broken lines); i.e. the H…F and H…O contacts corresponding to the O–H…F and C–H…O hydrogen bonds, respectively. The isolines of the laplacian of electron density, ∇ρ²(r), are also presented in this figure and it is indicated where the laplacian value is negative or positive. The latter indication is useful to describe different characteristics of the system analyzed such as the Lewis acid and Lewis base sites, the nature of interatomic contacts (bonds or intermolecular contacts), the regions of the concentration and depletion of electron charge density—the latter often leads to the location of unshared electron pairs, to the identification if the bond is ionic or covalent in nature, etc. [13–15]. For numerous molecular graphs analyzed hereafter these laplacian isolines or the reactive surfaces (∇ρ²(r) = 0 isosurfaces) are also presented to deepen the understanding of the nature of interactions.

Fig. 15.1

The molecular graph of the C₆FH₅–H₂O complex, solid and broken lines correspond to bond paths, big circles to attractors and small circles to critical points (green–bond critical points, red–ring critical points), the isolines of laplacian of electron density are also presented; positive values are depicted in solid lines and negative values in broken lines (this is the rule for the other figures of this chapter); the laplacian isodensity lines in the plane of the complex; based on the MP2/aug-cc-pVTZ results of calculations

There is no difference if, from classical point of view, the intermolecular interaction or the chemical bond is considered. For both cases the bond path is created what means that two atoms are bonded (but it does not mean that necessarily there is the bond between them). Sometimes the characteristics of the bond critical point (BCP) related to the bond path considered are analyzed, the negative value of the laplacian of the electron density at BCP, ∇²ρ_BCP, indicates the concentration of the electron charge density in the inter-atomic region what is typical for covalent bonds. If the positive value of ∇²ρ_BCP is observed thus there is the closed-shell interaction like in a case of an ionic bond, a van der Waals interaction or a hydrogen bond. However the above classification is not always a rule. Sometimes for strong hydrogen bonds the total electron energy density at BCP, H_BCP, is negative or even for very strong hydrogen bonds the ∇²ρ_BCP value is negative like for the typical covalent bonds [16, 17]. It was stated that the interaction is covalent in nature or at least it is characterized by the partial covalency if H_BCP for the analyzed interatomic contact is negative, there is no requirement of the negative value of ∇²ρ_BCP [18, 19]. One can mention here well known relationships between energetic topological parameters and the laplacian of the electron density at BCP (expressed in atomic units, see Eq. 15.1) [5, 6].

$${1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}\nabla^{ 2}\uprho_{\text{BCP}} = 2 {\text{G}}_{\text{BCP}} + {\text{V}}_{\text{BCP}} ,\quad {\text{where}}\,\, {\text{H}}_{\text{BCP}} = {\text{V}}_{\text{BCP}} + {\text{G}}_{\text{BCP}}$$

(15.1)

G_BCP and V_BCP are the components of the mentioned above H_BCP energy density and represent the kinetic electron energy density and the potential electron energy density, respectively. G_BCP is always a positive value while V_BCP is always negative.

For the C₆FH₅–H₂O complex presented here the typical medium in strength or weak hydrogen bonds are created where both H_BCP and ∇²ρ_BCP values are positive. The electron density at the H…F BCP, ρ_BCP, corresponding to the O–H…F hydrogen bond is equal to 0.014 au while the ρ_BCP value for the H…O contact of the C–H…O hydrogen bond is equal to 0.009 au. The electron density at BCP corresponding to intermolecular contact, especially in a case of hydrogen bonds, is often treated as the measure of the strength of interaction [16, 17]. The presented values of ρ_BCP indicate that the hydrogen bonds considered here are rather weak, for example the ρ_BCP for the H…O intermolecular contact for water dimer linked through the medium in strength O–H…O hydrogen bond amounts ~0.02 au (this value depends slightly on a level of calculations) [20].

It was pointed out that the n(B) → σ ^*_AH orbital-orbital interaction may be treated as a signature of the A–H…B hydrogen bond since it corresponds to the H…B contact [21–24]. For the hydrogen bonds the latter orbital-orbital interaction energy is the most important contribution to the charge transfer energy term. For two O–H…F and C–H…O hydrogen bonds presented in Fig. 15.1 there are the n(F) → σ ^*_OH and n(O) → σ ^*_CH orbital-orbital interactions, respectively, with the corresponding energies amounting 1.7 and 0.8 kcal/mol (HF/aug-cc-pVTZ//MP2/aug-cc-pVTZ level). The corresponding n(O) → σ ^*_OH orbital-orbital interaction for the mentioned above water dimer is equal to ~6 kcal/mol (depending on the level of calculation).

One can see two ring critical points (RCPs, red small circles in Fig. 15.1) attributed to the benzene ring and to the ring created by covalent bonds and two H…F and H…O contacts; i.e. the latter RCP corresponds to the O–H…F–C–C–H…O ring. Figure 15.1 shows also the isolines of the laplacian of electron density with the regions of the concentration of the electron density at O and F proton acceptor centers. Those regions correspond to the lone electron pairs which are responsible for the existence of the negative electrostatic potentials (EPs) at the oxygen and fluorine atoms. The hydrogen atoms of the (C)H…O and (O)H…F contacts are characterized by the positive EPs due to the significant outflow of the electron density from the hydrogen atoms to the carbon and oxygen atoms and to the C–H and O–H inter-atomic regions. Figure 15.2 presents maps of the electrostatic potential for the H₂O and C₆H₅F molecules calculated separately at 0.001 au electron density molecular surfaces. The distributions of the EP for both species are in line with the observations presented earlier here and concerning the electron charge distribution.

Fig. 15.2

The maps of the electrostatic potential calculated at the 0.001 au molecular electron density surfaces for H₂O (left) and C₆H₅F (right) molecules; red and blue colors correspond to negative and positive EP, respectively. The maximum and minimum EP-values for H₂O are equal to +0.0709 au (H-atoms) and −0.0514 au (O-atom), respectively; such values for C₆H₅F are equal to +0.0296 au (H-atoms) and −0.0259 au (F-atom); MP2/aug-cc-pVTZ level of calculations

There is another interesting observation for the C₆H₅F–H₂O complex (Fig. 15.1). For all O–H, C–H and C–C covalent bonds one can see the continuous regions of the negative laplacian surrounding the corresponding bond critical points; with one exception of the C–F bond where there is the region of positive laplacian between two corresponding nuclei; even the positive ∇²ρ_BCP value is detected for the C–F bond critical point. This may indicate that the C–F bond is mostly ionic in nature and that there is the polarization of C–F bond with the concentration of the electron charge density at the fluorine centre. The polarization of C–F bond (the percentage of the electron charge density calculated at F-centre) evaluated within NBO approach is equal to 72.8 %. The QTAIM integrated charges of carbon and fluorine in this bond are equal to +0.472 au and −0.714 au, respectively.

The aim of this chapter is to show, on the basis of several examples, how the location of the bond path may be useful to characterize, define and/or verify the specific, considered interaction. Mainly the QTAIM approach [4–7] is considered here; however sometimes there are also references to other methods and concepts as for example; the Natural Bond Orbitals (NBO) method [21, 22] or the σ-hole concept [25–27]. This is worth to note that the results presented hereafter are mainly based on the MP2/aug-cc-pVTZ level of calculations; those results are taken from earlier studies or the calculations were carried out especially for the purposes of this chapter. Consequently the QTAIM calculations were performed on the MP2/aug-cc-pVTZ wave functions. The binding energies (E_bin’s) were calculated as differences between the energy of the complex and the sum of energies of monomers optimized separately and they were corrected for the basis set superposition error (BSSE) by the counterpoise method [28]. Since the NBO method is based on the Hartree-Fock method thus the corresponding NBO results, i.e. orbital-orbital interactions or atomic charges, if presented, are based on the HF/aug-cc-pVTZ//MP2/aug-cc-pVTZ level. Hence there is rather not indicated the level of calculations for the next systems discussed hereafter; unless the results presented were obtained within other levels of calculations.

2 The Case of Halogen Bond

The halogen bond interaction is one of the most interesting phenomena analyzed during the last few decades. It was found that the halogen atoms (designated later here as X), especially if connected with carbon (in C–X bonds), often play a role of electron acceptors (Lewis acid centers) interacting with the Lewis bases, i.e. with the electron rich species. This seems to be strange since halogen atoms are commonly known as the electronegative centers. There were various trials to explain this phenomenon. One can mention concepts of the anisotropy of the van der Waals radii of halogens [29] or of the anisotropy of the electron charge distribution around the halogen atoms [30]. According to the latter concept the electron density distribution of the halogen atom is characterized approximately by the ellipsoid shape, the major axis of the ellipsoid is perpendicular to the C–X bond while the minor axis belongs to the C–X bond line. This is why the X-centre interacts with nucleophiles in the direction being the elongation of the C–X bond while electrophiles interact with halogens in the direction perpendicular to the C–X bond or nearly so. The latter direction is rich in electron density while the elongation of the C–X bond is characterized by the deficiency of the electron density. These explanations are in line with experimental findings; especially those based on the crystal structures determinations [31].

It seems that more recent σ-hole concept [25–27], being in line with explanations based on the anisotropy of the electron charge distribution, explains additionally the source of such anisotropy as well as this concept explains sufficiently the phenomena of numerous other interactions, not only of the halogen bond. The σ-hole concept of Clark, Murray and Politzer is based on simple models of electron configurations and hybridizations [32, 33]. For example, in a case of the CF₃X molecules (X = Cl, Br and I) the approximate s²p ²_x p ²_y p ¹_z configuration is observed for X-atoms, where the Z-axis is along the C–X bond. The unshared electron pairs are responsible for the existence of the negative electrostatic potential (EP) around the X-atom in the direction approximately perpendicular to the C–X bond while the single p ¹_z electron is involved in the C–X σ-bond what results in the loss of the electron density on the outermost portion of the halogen surface, in the elongation of the C–X bond. This is why this region (σ-hole) is characterized by the positive EP.

Figure 15.3 presents the map of EP for the CF₃Cl molecule. One can see here regions of the negative EP attributed to the fluorine atoms (red color in Fig. 15.3) as well as less negative region of EP around the Cl-atom (orange color in Fig. 15.3) being the consequence of the existence of unshared electron pairs. The positive region of EP in the elongation of the C–Cl bond is observed (blue color). This is interesting that there are also regions of the positive EP in the elongation of F–C and Cl–C bonds, attributed to the carbon which may also act as the Lewis acid sites. Such regions are also nicely explained in terms of the σ-hole concept.

Fig. 15.3

The map of the electrostatic potential calculated at the 0.001 au molecular electron density surface for CF₃Cl molecule; red and blue colors correspond to negative and positive EP, respectively

In general the σ-hole concept explains satisfactory why the halogen centers possess the dual character; they may act as the Lewis acid and as the Lewis base. In the former case the corresponding interaction is named as the halogen bond and it is usually linear or nearly so due to the location of the restricted area of the positive EP in the elongation of C–X bond.

Different complexes of the CF₃Cl moiety were analyzed recently and the dual character of the chlorine centre was also investigated. Figure 15.4 shows the molecular graph of the CF₃Cl…Cl⁻ complex where the chlorine anion is directed to the positive EP region of the Cl-centre in the CF₃Cl molecule. The linear C–Cl…Cl⁻ link is observed what is in line with the σ-hole concept and the earlier explanations presented here. The bond path linking chlorines with the corresponding BCP is observed. The isolines of laplacian of the electron density are also presented showing the anisotropic concentration of the electron density around the (C)Cl chlorine centre while the unperturbed spherical concentration of the electron density is observed for chlorine anion. One can also see the regions corresponding to other σ-holes, attributed to the carbon, i.e. to the elongations of the F–C and Cl–C bonds.

Fig. 15.4

The molecular graph of the CF₃Cl…Cl⁻ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of laplacian of electron density are also presented; the laplacian isodensity lines in the plane containing CFCl atoms of the CF₃Cl molecule and Cl⁻ anion

Figure 15.5 presents the molecular graph of the CF₃Cl–HF complex where the positively charged and characterized by the positive EP hydrogen atom of HF is directed to the “belt” of the negative EP around chlorine atom; this “belt” is visible from the map of the laplacian isolines performed in the plane containing CFCl atoms of the CF₃Cl molecule. In other words the Cl-centre plays the role of the Lewis base in this complex. There are other interesting findings here. The HF molecule is situated outside of the mentioned above CFCl plane. This is why the 2D map with laplacian isolines do not show the negative regions of the laplacian for the HF molecule since the latter species is situated outside the molecular graph projection. Probably such a configuration of the complex is observed because of the repulsion between the fluorine atoms of CF₃Cl and the fluorine atom of HF molecule. Figure 15.5 presents also the reactive surface (∇²ρ(r) = 0 isosurface); this picture may be read in the following way, the areas closed by those lines are characterized by the negative values of the laplacian of electron density thus they correspond to the concentration of the electron charge.

Fig. 15.5

The molecular graph of the CF₃Cl…HF complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of laplacian of electron density are also presented (the laplacian isodensity lines in the plane containing CFCl atoms of the CF₃Cl molecule); the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented in the right part of the figure

It was explained in terms of the σ-hole concept that the fluorine usually does not possess the regions of positive EP since the high electronegativity of F-centre as well as the sp-hybridization cause that the σ-hole usually occurring in the outermost part of C–F bond is neutralized [32, 33]. Additionally the Lewis acid strength for halogen atoms increases in the following order, F < Cl < Br < I what is connected with the increase of the polarization of the corresponding C–X bonds (% of the electron density calculated at the carbon atom increases). For example, the HCCX halogen derivatives of acetylene were analyzed and it was found that the maximum EP at X center is equal to +2.3, +20.4 and +27.0 kcal/mol for X = F, Cl and Br, respectively (B3PW92/6-31G(d,p) level of calculations) [34]. There are similar findings of an increase of the positive EP for heavier X-center for other species, like for example H₃CX, F₃CX or C₆H₅X [34].

Figure 15.6 presents the molecular graphs of HCCCl and HCCBr molecules with the isolines of the laplacian of electron density. One can see the anisotropic concentration of the electron charge density around chlorine for the HCCCl molecule, with its “thin” concentration in the elongation of C–Cl bond and wider concentration in the direction perpendicular to this bond; the latter corresponds to the lone unshared electron pairs. In a case of HCCBr molecule there is no the concentration of the electron density in the C–Br bond line! This is why the more positive EP is observed for the bromine derivative of acetylene than for the chlorine derivative.

Fig. 15.6

The molecular graphs of the HCCCl (up) and HCCBr (down) molecule, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of Laplacian of electron density are also presented

In a case of halogen bond the monovalent halogen center is usually analyzed while there are only few studies on that kind of interaction with multivalent halogen atoms. One can mention the study on Ph₂IX complexes with XF₃ (X = Cl, Br, I) [35] or the study where the complexes of XF₃ are compared with their XF analogues (X = Br or Cl) [36]. Very recently the complexes of BrF₃ and BrF₅ with N₂ and HCN species acting as Lewis bases were analyzed [37]. The situation for the bromine center in BrF₃ and BrF₅ is different than in a case of monovalent halogens. First of all, the whole hemispheres of multivalent bromines are characterized by the positive electrostatic potential. It means that bromine should play the role of the Lewis acid only and not of the Lewis base. For the BrF₃ moiety characterized by the C_2v symmetry the maximum positive EP occurs for the Br center in the elongation of the F–Br equatorial bond (Fig. 15.7). Slightly different situation is observed for the C_4v symmetry BrF₅ molecule where four equivalent maxima of EP are observed at Br-center, around the fourfold symmetry axis of the molecule (Fig. 15.7).

Fig. 15.7

The map of the electrostatic potential calculated at the 0.001 au molecular electron density surfaces for BrF₃ (left) and BrF₅ (right) molecules; red and blue colors correspond to negative and positive EP, respectively. Black points designate the maxima of EP, one local maximum for BrF₃ and 4 local maxima for BrF₅ (only 3 maxima are visible in the figure)

The distribution of the electrostatic potential for two bromine species has further consequences; particularly the location of the maxima of EP shows the most probable nucleophilic attacks here. This is why the liner halogen bonds are formed with BrF₃ while bent ones for the BrF₅ molecule. Figure 15.8 shows the molecular graph of the BrF₃–NCH complex with the reactive surface corresponding to the laplacian of the electron density equal to zero. The straight bond path connecting the bromine atom with the nitrogen Lewis base center of HCN molecule is observed here; one can say that the bond path crosses the maximum of EP of the bromine atom. It is worth to mention that the BrF₃ molecule displays a T-shaped molecular structure and a trigonal bipyramidal electronic structure. Two lone unshared electron pairs are located symmetrically to the mirror plane containing the BrF₃ molecule. This picture is in line with the chemical intuition and with the NBO results. The latter approach confirms the existence of two unshared electron pairs for bromine which is also characterized by 14 core orbitals (28 electrons); 3 remaining bromine electrons are involved in Br–F bonds. The electron configurations for bromine in the isolated BrF₃ molecule and in the BrF₃–NCH complex are practically the same according to the NBO approach, only there are negligible differences in the occupancies of orbitals.

Fig. 15.8

The molecular graph of the BrF₃…NCH complex, solid lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is also presented

As it was pointed out earlier here, the different situation is observed for the BrF₅ moiety. For example, for the BrF₅–N₂ complex the Lewis base N-center of molecular nitrogen is directed to one of the BrF₅ EP maxima. Figure 15.9 presents the corresponding molecular graph of that complex with the Br…N bond path crossing the local EP maximum and partly avoiding the unshared electron pair of bromine. The bromine electron configuration derived from NBO for the BrF₅ moiety is as follows, 28 core electrons, 1 lone electron pair and 5 remaining electrons are involved in 5 F–Br σ-bonds. This means that for the square pyramidal BrF₅ molecular structure observed here there are 12 electrons in the valence shell. The latter is often named as hypervalency in the literature [38] since the octet rule is not obeyed here. The electron configuration of bromine in the BrF₅–N₂ complex is practically the same as in the BrF₅ moiety not involved in any interaction, similarly as it was observed in a case of the BrF₃ species and its complexes.

Fig. 15.9

The molecular graph of the BrF₅…N₂ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

Very interesting situation is observed for the BrF₅–Cl⁻ complex (Fig. 15.10). This complex is characterized by a very strong interaction since the binding energy amounts here −37.8 kcal/mol while for the complexes of BrF₅ with HCN and N₂ the binding energy is equal to −5.1 and −1.7 kcal/mol, respectively [37]. The laplacian of the electron density, ∇²ρ_BCP, at Br–Cl BCP is positive; however the total electron energy density at BCP, H_BCP, is negative what means that the interaction is at least partly covalent in nature. The similar situation is observed for all five Br–F connections—the positive ∇²ρ_BCP values and the negative H_BCP ones. It seems that the strong interaction of the BrF₅ species with the Cl⁻ anion extorts the greater electron density changes in the complex, particularly in the BF₅ moiety. One lone electron pair of bromine is observed in the complex within the NBO approach, similarly as in the isolated BrF₅ species; however the lone pairs in the complex possesses s-character in 99.2 %. This is confirmed by the QTAIM approach since Fig. 15.10 shows the spherical electron charge density concentration around bromine center. The change of the location of the bromine electron lone pair results in the existence of the straight bond path linking the Br and Cl centers in contrary to the interactions of the BrF₅ molecule with weak Lewis bases (see Fig. 15.9).

Fig. 15.10

The molecular graph of the BrF₅…Cl⁻ complex, solid lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

The positive values of the laplacian of electron density at all BCPs corresponding to the Br–F and Br–Cl links in the BrF₅–Cl⁻ complex may indicate the ionic character of interactions since it was pointed out that such values are typical for the closed-shell interactions [5–7]. This means that in the latter complex all bonds (links) are strongly polarized; the NBO approach shows that the polarization for the Br–F bonds is equal to 23.2 % and it amounts 33.6 % for the Br–Cl bond (% of the electron density at the Br center). The Br atomic charge calculated within NBO and QTAIM approaches is equal to +2.25 au and +2.32 au, respectively. Figure 15.10 clearly shows the concentration of the electron density at attractors and not at interatomic regions; the similar situation is observed for all BrF₃ and BrF₅ complexes.

There is another interesting finding for the BrF₅-Cl⁻ complex; the NBO approach shows that there are tri-center four-electron (3c-4e) linear bonds here. They concern the following linear triads in the BF₅Cl⁻ moiety; two F–Br–F triads and one F–Br–Cl triad. Such 3c-4e bonds were analyzed earlier in literature; for example Weinhold and Landis analyzed numerous hypervalent centers [21] and they extended the 3c-4e concept of Pimentel and Rundle [39, 40].

3 Lewis Acid–Lewis Base Interactions

The σ-hole concept mentioned in the previous section which explains the mechanisms of formation of the halogen bond is also useful to analyze other Lewis acid–Lewis base interactions [26, 27, 41]. One can mention tetrel [42–45], pnicogen [46–50] and chalcogen bonds [51–53] where the elements of Groups IV, V and VI, respectively, play the role of the Lewis acid centers. The mechanism of the formation of those bonds is practically the same as that one of the halogen bond; a region of positive electrostatic potential (designated as σ-hole) on the extension of the bond to the atom being the acidic center is an effect of the electron charge shift from the outermost part of this center. These centers possessing σ-holes are usually characterized by the unshared electron pairs thus they may act simultaneously as the Lewis acids and the Lewis bases. The situation is slightly different for the tetrel atoms which are characterized by the sp³ hybridization and do not possess free electron pairs. That rather excludes their potential basicity. Similarly, as it was mentioned earlier here in the previous section, multivalent halogens may be characterized only by acidity properties but in that case the halogen centers possess lone electron pairs.

The dual character mentioned few times here results in important consequences that the same elements may be involved in the stabilization interaction; i.e. that halogen-halogen, chalcogen-chalcogen, etc. interactions are possible. That is not all since at least two different in nature connections may be mentioned here. For example, the X halogen center may interact by its Lewis acid region (σ-hole) with the Lewis base region (unshared electron pairs) of the X-center of another molecule or both halogen centers being in the X…X contact may play the dual role, both may act simultaneously as Lewis acid and Lewis base. Both cases of C–X…X–C halogen-halogen interactions were described in early study of Fourmigue and Batali [31] and they are presented in Scheme 15.1. In a case of the interaction designated as I, the halogen atoms act simultaneously as Lewis acid and Lewis base centers. Such centers may be sometimes related by symmetry, for example, in the crystal structures such halogen atoms may belong to symmetry equivalent molecules. In a case of the type II interaction one of halogens acts as the Lewis base and the second one as the Lewis acid in spite of the fact that potentially both possess dual character.

Scheme 15.1

Two types of C-X…X–C halogen-halogen interactions; this figure is based on the scheme presented in Ref. [31]; the ellipsoids represent here the non-spherical distribution of the electron charge density

Similar types of interactions are observed for other σ-hole bonds. For example, P…P pnicogen interactions were analyzed in numerous complexes [54, 55]. For the FH₂P–PH₃ complex the σ-hole of the FH₂P moiety situated in the elongation of the F–P bond interacts with the lone electron pair of the PH₃ species; this corresponds to the n(P) → σ ^*_PF orbital-orbital interaction within the NBO approach; in other words the FH₂P species reveals here the Lewis acid properties while the PH₃ molecule the Lewis base properties. One can expect the reverse situation, where the FH₂P and PH₃ molecules act as the Lewis base and the Lewis acid, respectively with the corresponding n(P) → σ ^*_PH orbital-orbital interaction. However the P center possesses the stronger Lewis acid properties in the FH₂P molecule than in the PH₃ species since the n(P) → σ ^*_PF and n(P) → σ ^*_PH energies of interactions in the FH₂P–PH₃ complex are equal to 8.9 and 2.7 kcal/mol, respectively. Besides the FH₂P σ-hole (elongation of the F–P bond) is characterized by the more positive EP than the PH₃ σ-hole (elongation of H–P bond) [54, 55]. Figure 15.11 presents the molecular graph of this complex with the reactive surface (∇ρ²(r) = 0 isosurface). One can see that two molecules of the complex are oriented in such a way that lone electron pair (the electron charge density concentration) is directed to the σ-hole region.

Fig. 15.11

The molecular graph of the FH₂P…PH₃ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

On the other hand the (FH₂P)₂ dimer represents the situation of the equivalent phosphorus atoms, both acting simultaneously as the Lewis acid and as the Lewis base what corresponds to two equivalent n(P) → σ ^*_PF orbital-orbital interactions (each characterized by the energy of 29.5 kcal/mol). Figure 15.12 presents the molecular graph of this complex. The authors described also P…P interactions in other complexes [55]; for example for the FH₂P…P(OH)H₂ complex there are two important orbital-orbital interactions: n(P) → σ ^*_PF and n(P) → σ ^*_P–OH ; however the former one is stronger than the latter one what means that the P center is more acidic in the PFH₂ moiety than in the P(OH)H₂ one.

Fig. 15.12

The molecular graph of the (FH₂P)₂ dimer, solid lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

The results presented here show that for the pnicogen-pnicogen contacts there is no clear border between type I and II configurations as it was found for halogen-halogen interactions (Scheme 15.1). It seems that an orientation corresponding exactly to the II type orientation found for the X…X contacts cannot be formed for the P…P interaction. It probably comes from the existence of three σ-holes for trivalent P centers while in a case of monovalent halogens there is only one σ-hole characterized by the acidic properties.

This is worth mentioning that in both cases of pnicogen-pnicogen interaction there is only one bond path linking the corresponding attractors (see Figs. 15.11 and 15.12). It seems that this situation may be generalized for interactions between the same kind centers, i.e. for the chalcogen-chalcogen or halogen-halogen (dihalogen) bonds. The results presented here show that the molecular graph does not reflect more complex situation, i.e. differences between those two cases. Such difference is reflected within the NBO analysis. However it seems that the QTAIM approach provides the tools to describe that more complex situation. The σ-hole bonds, i.e. halogen and pnicogen bonds, were analyzed recently by Eskandari and co-workers [56, 57]. They have shown that the P…P, P…N and N…N interactions [56], as well as halogen bonds [57], may be categorized as lump-hole interactions where the hole is a region of charge depletion and the excess of the kinetic energy while the lump is a region of the electron charge concentration and an excess of the potential energy. In such a way the authors have found one hole-lump interaction for the FH₂P…PH₃ complex and two equivalent hole-lump interactions for the (FH₂P)₂ dimer [56] what corresponds to one dominant or two equivalent orbital-orbital interactions, respectively.

There is a very interesting case of tetrel bonds where the Group IV element plays the role of the Lewis acid center. The tetrel bonds were analyzed recently as a kind of σ-hole bonds [42–45]; however such interactions were known early on as the preliminary stages of the S_N2 reactions [58, 59]. For example, the methane species may be treated as a very weak acid interacting through the H-atom with the Lewis base centre; H-atoms of methane are characterized by the positive EP (+0.015 au at 0.001 au molecular surface) while C-centre by slightly negative EP (−0.004 au). Figure 15.13 presents the molecular graph of the CH₄…NCH complex corresponding to this situation where the H…N bond path corresponding to the C–H…N hydrogen bond is observed. However for the CH₃Cl derivative there is the region of the positive EP (+0.032 au) at the C-center in the elongation of the Cl–C bond, this is the σ-hole which plays the role of the Lewis acid in interactions with electron rich species. Figure 15.14 presents the molecular graph of such situation; this is the CH₃Cl…F⁻ complex which may be treated as the preliminary stage of the CH₃Cl + F⁻ → CH₃F + Cl⁻ reaction governed by the S_N2 mechanism.

Fig. 15.13

The molecular graph of the CH₄…NCH complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs

Fig. 15.14

The molecular graph of the CH₃Cl…F⁻ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs

Let us look at few experimental studies; the addition reactions of methyl(halogeno)tin and methyl(halogeno)germanium compounds to electron-rich platinium(II) complexes were analyzed [60]. These studies clearly point to an S_N2 mechanism for the interactions considered. It was found that methylhalotin species, Me₃SnX (X = Cl, Br, I), react by an S_N2 pathway. The germanium analogues were also analyzed. For example, the variable temperature NMR spectral series for Me₃GeCl shows the rate decrease of the S_N2 reaction in comparison with the analogues tin species and the authors explain that this is partly due to a size effect. This is in line with the σ-hole concept as well as with the topology of complexes linked through the tetrel bonds. It was found that for the tetrel atoms as well as for the other Groups centers the positive EP increases in the Group if the atomic number of the center increases. For example, for the CH₃F, SiH₃F and GeH₃F series the maximum EP at tetrel atom, in the elongation of fluorine—tetrel center bond, is equal to +0.033, +0.062 and +0.068 au, respectively [45]. Hence the S_N2 reaction mentioned earlier here and preceded by the tetrel bond is more probable for the tin species than for the germanium one. There is also the influence of the size effect. It was claimed that the methyl groups are more important to block the tetrel center if the latter one is characterized by the smaller size [60]. Figure 15.15 presents a similar situation of the influence of fluorine substituents for the CF₄…Cl⁻ and GeF₄…Cl⁻ complexes. For the carbon species the Cl-attractor is linked by the bond paths with the fluorine attractors of CF₄ species; in a case of the germanium complex there is the Cl…Ge intermolecular bond path what means that for the germanium complex the S_N2 reaction is more probable.

Fig. 15.15

The molecular graphs of the CF₄…Cl⁻ (left) and GeF₄… Cl⁻ (right) complexes, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs (red ones—RCPs), the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

The similar situation occurs for ammonium cation, NH₄ ⁺ and its analogues, PH₄ ⁺ and AsH₄ ⁺ [50]. However for those cations the whole molecular surfaces are characterized by the positive EP thus they do not act as Lewis bases. For the NH₄ ⁺ ion the maximum EP amounts +0.286 au and it is attributed to the H-atoms while the minimum to the nitrogen center, +0.263 au. This is why for the complexes with hydrogen cyanide (through N-center) as well as with the other Lewis base centers [61], firstly the H-atoms of ammonia cation are involved in interactions and next the links are formed with the nitrogen center of NH₄ ⁺. For example, for the NH₄ ⁺…(NCH)_n clusters, the N–H…N hydrogen bonds are formed for n ≤ 4 and next, when all H-atoms are saturated by the hydrogen bonds the N…N links are created. The latter ones correspond to the σ-hole bonds. Figure 15.16 presents the NH₄ ⁺…(NCH)₅ cluster where one can observe four N–H…N hydrogen bonds and one N…N interaction; for all interactions the corresponding bond paths exist.

Fig. 15.16

The molecular graph of the NH₄ ⁺…(NCH)₅ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

4 The Case of Hydrogen Bond

It was discussed that the A–H…B hydrogen bond may be classified as a case of the σ-hole bond [33, 62]. For the A–H bond, similarly as for other covalent bonds where the σ-hole is observed, there is the electron charge density shift from H-atom to the A-center. However the mechanism of the σ-hole formation is different for the H-atom than for the other heavier atoms [33, 63]. In a case of pnicogen, chalcogen and halogen atoms p-electron is involved in the σ-bond what results in the depletion of the electron density in outermost part of the center considered and a small area of the increased electrostatic potential, often up to the positive value [26, 27]. Such a region of the positive EP, if exists, is surrounded by areas of the negative EP. There are exceptions, as for example for the tetrel atoms or multivalent halogens described earlier here, where the whole centers are characterized by the positive EP. For the H-atom the whole hemisphere is characterized by the positive EP and there is the single s-type electron involved in the A–H bond, the polarization of this bond results in the additional increase of the EP at the hydrogen atom.

This is interesting that for the protic H-atom in A–H bond there is the detectable shift of its electron density maximum towards the A-center. This effect is not observed for halogen, chalcogen or any others heavier atoms. That may be calculated within the QTAIM approach since the positions of attractors correspond to the positions of the local maxima of the electron density; they may be attributed to the positions of atoms. However if one assumes that the atoms’ positions correspond to nuclei thus this is not the same. Table 15.1 presents MP2/aug-cc-pVTZ results for the selected molecules and cations containing hydrogen atoms. One can see that for all systems considered, for A–H bonds (A = C, N, O, F, Si, P, S, Cl) the distance between the attractors (r) is shorter than the distance between corresponding nuclei (R). Since the positions of attractors and nuclei for heavy (non-hydrogen) atoms are practically the same thus the R-r difference is exactly the shift of the electron density maximum of H-atom from the corresponding nucleus to the A-atom. The R-r difference may be also read as the distance between the nucleus of H-atom and its attractor. The range of those differences is between 0.01 Å and 0.03 Å (Table 15.1), the greatest values are observed for the HF molecule and H₃O⁺ cation, 0.034 Å and 0.032 Å, respectively. The ΔR% values relate those differences to the R distances (between nuclei) since ΔR% = [(R-r)/R)100 %].

Table 15.1 A–H bonds of selected molecules and cations (A designates heavy atom); r—the distance between A and H attractors; R—the distance between A and H nuclei; ΔR% is equal to [(R-r)/R]*100 %; ε—electronegativity of the A heavy atom in the Pauling scale; distances in Å

It seems that the greatest R-r differences occur for the more electronegative A-atoms; this seems to be natural since the more electronegative center withdraws easier the electron density. Figure 15.17 presents the dependence between the electron charge density shift (expressed as the ΔR% value) and the electronegativity of the non-hydrogen A-atom. The results for molecules and cations are included. One can see correlations within the periods, i.e. if the A-atom belongs to the same period and only the neutral species are considered. In a case of cations there is the greater electron charge density shift if one compares it with the neutral molecules. For example, the R-r value for the NH₃ molecule is equal to 0.022 Å while for the NH₄ ⁺ cation it amounts 0.025 Å; for H₂O and H₃O⁺ species it is equal to 0.027 Å and 0.032 Å, respectively.

Fig. 15.17

The relationship between ΔR% value (for species collected in Table 15.1) and the electronegativity of the A non-hydrogen atom, circles correspond to neutral molecules while crosses to cations

Figure 15.18 presents the O–H bond length histograms based on results from Cambridge Structural Database (CSD) [64]; X-ray and neutron diffraction results are presented separately. The CSD searches were performed for all derivatives of benzene containing OH group, this means that not only phenol is considered but also species where other H-atoms of benzene are substituted. Of course such substituents may influence on the electron density distribution including the hydroxyl group. However the aim of those searches was only to evaluate briefly the differences between the neutron and X-ray diffraction results if the covalent bond containing H-atom is considered. In a case of the neutron diffraction the positions of atoms in the crystal correspond to the positions of nuclei while for the X-ray measurements these are the positions of the local maxima of electron density; in other words in the latter case we have the positions of attractors. This is why the QTAIM approach may be applied for the crystal structures’ results [1–3] and for numerous crystal structures the topological analysis based on the experimental electron density is performed. The mentioned above searches on the phenol derivatives were carried out for high quality measurements since the following search criteria were applied: no disordered structures, no structures with unresolved errors, no powder structures, no polymeric structures, e.s.d.’s for C–C bonds ≤ 0.005 Å and the discrepancy index R ≤ 7 %.

Fig. 15.18

Histograms of the O-H bond length for O–H groups connected with benzene ring detected in crystal structures, neutron diffraction (down) and X-ray diffraction results (up) are presented. The horizontal axes correspond to O–H bonds (length in Å), the vertical axes to number of observations, based on the results from the Cambridge Structural Database [64]

The precision of the determination of the positions of the maxima of electron density for H-atoms in X-ray diffraction measurements is much lower than the precision of determination of hydrogen nuclei positions in the neutron diffraction [8] but one can see that statistically the X-ray O–H bonds are much shorter than the neutron diffraction counterparts (Fig. 15.18); the mean OH bond length for X-ray experiments is equal to 0.871 Å while this value for neutron diffraction amounts 0.990 Å; the corresponding median values are equal to 0.841 Å and 0.984 Å, respectively. The median values are presented here since the bond length distributions are not the normal ones. The number of the OH bonds included in the histograms (Fig. 15.18) for the X-ray and neutron diffraction is equal to 3798 and 12, respectively. This is because, in general, the number of X-ray observations collected in CSD is large in comparison with the neutron diffraction ones; the total number of crystal structures stored into Cambridge Structural Database is equal to 686 944 while only 1616 of them are the neutron diffraction measurements (CSD summary statistics—6 January 2014 [65]). It is worth mentioning that the total number of crystal structures deposited in CSD at the end of 2014 reached 750 200. In spite of the great disproportion between the number of neutron diffraction and X-ray results one can see that the experimental results clearly show that there is the electron charge density shift from H-atom to the bonded center, in the case considered here (Fig. 15.18) to the oxygen atom.

In view of those results one may say that the positive EP at hydrogen atom is enhanced owing to the electron density shift and that the links of H-atom with Lewis base centers, i.e. hydrogen bonds, may be classified as the σ-hole bonds. An example of the complex connected through two hydrogen bonds, O–H…F and C–H…O, was considered in the introduction (see Fig. 15.1). These are the classical examples of 3c-4e (3 center—4 electron) A–H…B hydrogen bonds since there are A, H and B centers, 2 electrons of the A–H σ-bond and the lone electron pair of the proton acceptor (B). However the A–H…π interactions, where it is difficult to indicate the single B-center, are also classified as hydrogen bonds. It is commonly accepted that for those interactions π-electrons play the role of the Lewis base.

Figure 15.19 shows the molecular graphs of the C₂H₂…HF T-shaped complex; two graphs with the isolines of the laplacian of electron density corresponding to two levels of calculations (MP2/aug-cc-pVTZ and MP2/aug-cc-pVQZ) are presented. In both cases there is the bond path linking the H-atom attractor of HF molecule with the π-electron system of acetylene; in the case of aug-cc-pVTZ basis set there is the link between H-attractor and the non-nuclear attractor (NNA) situated between carbon atoms, in a case of the aug-cc-pVQZ basis set there is the hydrogen–bond critical point (H…BCP) link. Practically there is no the significant difference between those results, one may say that both NNA and BCP mimic the one-center π-electron Lewis base. This is worth to mention that such H…π bond paths for hydrogen bonded systems were analyzed before [66]. The characteristics of the BCP corresponding to the H…π bond path are practically the same for both levels (ρ_BCP and ∇²ρ_BCP are equal to 0.021 au and 0.049 au, respectively; the remaining characteristics differ only slightly). The characteristics of BCP situated between carbon atoms for the aug-cc-pVQZ basis set are as follows; ρ_BCP = 0.416 au and ∇²ρ_BCP = −1.274 au while for the aug-cc-pVTZ basis set there are two BCPs between carbon attractors for which ρ_BCP = 0.411 au and ∇²ρ_BCP = −1.278 au, there is also NNA situated between latter BCPs where ρ_BCP = 0.411 au and ∇²ρ_BCP = −1.370 au.

Fig. 15.19

The molecular graphs of the C₂H₂…HF complex, the MP2/aug-cc-pVTZ (left) and MP2/aug-cc-pVQZ (right) levels, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of Laplacian of electron density are also presented; there is non-nuclear attractor (small red circle) between carbon atoms (left) for the MP2/aug-cc-pVTZ level

The situation is more complicated if the multicenter (more than 2 centers) π-electron system plays the role of proton acceptor in hydrogen bond interaction. It is often stated in the literature that aromatic systems or other closed-ring π-electron systems may play the role of proton acceptors in hydrogen bonds [67]; such species as benzene, antracene, cyclopentadienyl anion and their derivatives as well as other moieties are often considered. This means that π-electrons as a whole are treated as the Lewis base center. The hydrogen bonds with the multicenter π-electron systems were analyzed recently (the calculations were performed at the MP2/6-311 ++G(d,p) level) and it was found that at least few sub-classes of the A–H…π hydrogen bonds may be considered [68]. Figure 15.20 presents the molecular graphs of the C₆H₆…HCCH, C₆H₆…HF and C₅H₅ ⁻…HF complexes. The binding energy for the C₆H₆-HCCH complex amounts −2.1 kcal/mol (BSSE correction included), the complex is characterized by the C_6v symmetry and the molecular graph (Fig. 15.20) reflects this symmetry since six H…C bond paths between the hydrogen of acetylene and the carbon attractors of benzene are observed. For the BCPs of those bond paths the following characteristics were found; ρ_BCP = 0.007 au and ∇²ρ_BCP = 0.023 au. One may say that the C–H…π hydrogen bond is classified here as rather weak interaction and that all C-atoms of the benzene ring are involved equivalently in that interaction. On the other hand one may say that there are six equivalent C–H…C interactions for the C₆H₆…HCCH complex (bifurcated hydrogen bond); this situation is different than for the C₂H₂-HF complex presented earlier here where the H…BCP (or H…NNA) bond path was observed and where the BCP (or NNA) of the triple C ≡ C bond mimics the one-center Lewis base. Since within the NBO approach the Lewis type structures are considered thus three π_CC → σ ^*_CH intermolecular orbital-orbital interactions were found here, each of the energy amounting 0.4 kcal/mol and each corresponding to the overlap of CC π-bond orbital of benzene and the antibonding CH orbital of acetylene. The high C_6v structure symmetry results in the same electron charge distribution symmetry thus as it was pointed out earlier here, there are six equivalent C…H bond paths between acetylene and benzene as well as the equivalent charges for benzene atoms are detected (NBO charges for C and H atoms are equal to −0.208 au and +0.209 au, respectively; the corresponding charges for isolated benzene are equal to −0.204 au and +0.204 au).

Fig. 15.20

The molecular graphs of the C₆H₆…HCCH, C₆H₆…HF and C₅H₅ ⁻…HF complexes (from left to right), the MP2/6-311 ++G(d,p) level, solid and broken lines correspond to bond paths, big circles to attractors and small circles to critical points (green—BCPs, red—RCPs)

For the C₆H₆…HF complex there is the H…BCP bond path between the HF molecule and the BCP of the CC bond of benzene; thus it is the similar situation to that one occurring for the C₂H₂…HF complex. The interaction is stronger here than for the complex of benzene with acetylene since the binding energy is equal to −3.3 kcal/mol; the following characteristics were found for the BCP of the mentioned above H…BCP bond path; ρ_BCP = 0.010 au and ∇²ρ_BCP = 0.031 au. One can see that the C_6v symmetry is broken here; two carbon atoms are distinguished since the BCP of the path between them mimics the one-center Lewis base for this complex. The charges of those carbon atoms in the C₆H₆–HF complex are the most negative ones since they are equal to −0.225 au while the remaining C-charges amount −0.204 au; besides the intermolecular H…C distances are smallest for these carbons, 2.503 Å, while the remaining H…C (HF-benzene) distances amount 2.724−2.929 Å. This means that the bond path between HF and C₆H₆ molecules shows the preferable H…C interactions. What is the reason of such an orientation of the HF molecule in the C₆H₆–HF complex? Probably for this structure the electrostatic interactions between benzene and HF are more attractive than in a case of the C_6v symmetry complex. However the difference is not significant; the MP2/aug-cc-pVTZ calculations show that the transition state C_6v symmetry C₆H₆–HF complex is higher in energy than the corresponding complex being in the energetic minimum only by 0.1 kcal/mol. The single π_CC → σ ^*_CH orbital-orbital interaction is observed for the C₆H₆–HF complex with the interaction energy of 1.1 kcal/mol; the CC bond π-orbital corresponds here to the CC BCP which is connected with the H-atom attractor of HF molecule (see Fig. 15.20).

Figure 15.20 presents also the C₅H₅ ⁻…HF complex where the F–H…π hydrogen bond is assisted by the negative charge. It results in the stronger C₅H₅ ⁻–HF interaction since the binding energy is equal to −16.3 kcal/mol. The molecular graph shows the single H…C bond path where the following characteristic of the corresponding BCP were calculated; ρ_BCP = 0.028 au and ∇²ρ_BCP = 0.056 au. The total electron energy density, H_BCP, at this bond critical point is equal to -0.002 au what may be explained by the partly covalent in nature corresponding H…C interaction. Also in this case, similarly as it was found for the C₆H₆–HF complex, the bond path shows the preferable H…C interaction since the charge of the corresponding C-atom of the C₅H₅ ⁻ anion is equal to −0.433 au while for the remaining carbons of the anion this charge is situated in a range between −0.350 au and −0.353 au. The H…C distance corresponding to the bond path is equal to 2.007 Å while the remaining distances amount 2.395–2.902 Å. This is worth to mention that for the C₅H₅ ⁻…HF complex a lone electron pair orbital within the NBO approach is detected for the carbon atom of C₅H₅ ⁻ and that this orbital is involved in the most important n(C) → σ ^*_CH orbital-orbital overlap for which the energy of interaction amounts 9.7 kcal/mol; this orbital is located at the C-atom connected by the bond path with HF molecule.

One can see that for the A–H…π hydrogen bonds presented here there are significant structural differences between them; those differences are reflected in the geometry as well as in the electron charge distributions. One can observe hydrogen bonds where the protic H-atom is linked with the Lewis base by atom-atom bond path (in C₅H₅ ⁻…HF), by the atom-BCP bond path (in C₆H₆…HF or C₂H₂…HF) and by the few equivalent atom-atom bond paths (in the C₆H₆…HCCH complex). In such a situation the question arises if the meaning of the term A–H…π hydrogen bonds should be revised. For example, maybe for the C₅H₅ ⁻…HF complex the term F–H…C hydrogen bond is more proper than the F–H…π one and maybe for the C₆H₆…HCCH complex the bifurcated hydrogen bond (six F–H…C intermolecular contacts) should be applied.

It is interesting that also σ-electrons may play a role of the proton acceptor in the hydrogen bonds; they are designated as the A–H…σ interactions. Such systems were analyzed theoretically for the FH…H₂, H₃O⁺…H₂, NH₄ ⁺…H₂ and other complexes where the molecular hydrogen acts as the Lewis base [69–72]. For all of them the situation is similar to that one occurring for the C₂H₂…HF complex; i.e. there is the bond path linking the hydrogen protic atom of the Lewis acid unit with the bond critical point of H–H molecular hydrogen bond. Figure 15.21 presents the molecular graph of the NH₄ ⁺…H₂ complex where one may observe this type of the bond path. One can see that for this complex the N–H bond of ammonia cation is located perpendicularly (or nearly so) to the H₂ molecule. Such T-shaped structures are observed for other complexes mentioned here, where the σ-electrons of the molecular hydrogen are the proton acceptor in hydrogen bonds [71]. For the NH₄ ⁺…H₂ complex the binding energy is equal to −2.3 kcal/mol (MP2/6-311 ++(3df,3pd) level, BSSE included), the ρ_BCP and ∇²ρ_BCP values for the BCP corresponding to the H…BCP bond path linking ammonia cation with H₂ molecule are equal to 0.013 au and 0.041 au, respectively. This is interesting that the most important orbital-orbital interaction corresponds here to the σ_H-H → σ ^*_N–H overlap where σ_H–H is the σ-bond orbital of molecular hydrogen while σ ^*_N–H is the antibonding N–H orbital of ammonia cation, the corresponding energy of interaction is equal to 3.6 kcal/mol (HF/6-311 ++G(d,p)//MP2/6-311 ++G(d,p) level).

Fig. 15.21

The molecular graph of the NH₄ ⁺…H₂ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of Laplacian of electron density are presented; the projection in the plane containing H₂ molecule and HNH atoms of NH₄ ⁺ cation

The existence of the A–H…σ hydrogen bonds was confirmed by experiment. For example, gas phase measurements of dipole moment and vibrational predissociation lifetimes as well as the determination of rotational constants performed for the F–H…H₂ complex [73] confirmed its T-shaped structure. The high-resolution infrared spectra were analyzed for the H₂–HF, D₂–HF and HD–HF complexes solvated in helium nanodroplets and also for those species the T-shaped structure was confirmed [74–76]. The infrared vibrational predissociation spectra measurements were carried out for the H₂–HCO⁺ complex linked through C–H…σ interaction, where the σ-electrons of the molecular hydrogen act as the Lewis base [77]. The authors did not consider the C–H…σ interaction as the hydrogen bond; however the results of experiment clearly show that this arrangement possesses characteristics typical for the latter kind of interaction.

Finally it is worth mentioning that the clusters of ammonia cation, NH₄ ⁺, with H₂ molecules—(NH₄ ⁺)–(H₂)_n (n up to 8) were calculated [69, 78]. It was also found that the whole molecular surface of ammonia cation (0.001 au electron density surface was considered) is characterized by the positive EP; however the EP maxima are attributed to H-atoms [50]. Hence the increase of the number of molecular hydrogen molecules surrounding ammonia cation results in the N–H…σ hydrogen bonds formation and when all N–H bonds are saturated by those interactions (for n = 4) thus next H₂ molecules (for clusters with n ≥ 5) are linked with the nitrogen center of the cation, however all H₂ molecules for those clusters act as Lewis bases through their σ-electrons. The situation is very similar to that one occurring for the (NH₄ ⁺)–(HCN)_n clusters [61] mentioned earlier here (see also Fig. 15.16).

This is interesting that the molecular hydrogen possesses the properties of Lewis base (σ-electrons) and of the Lewis acid what is reflected in the distribution of electrostatic potential which is negative along the H–H σ-bond and it is positive at the outermost parts of H-atoms [79]. One may expect that the interaction of the H₂ molecule with Lewis acids should result in its connection through H–H BCP while the interaction with Lewis bases should result in the connection of the H₂ species by one of its H-attractors. Figure 15.22 presents the latter situation where H₂ interacts as the Lewis acid with the hydride Lewis base center of LiH species. Figure 15.22 clearly shows the H…H bond path linking basic (of LiH) and acidic (of H₂) hydrogen atoms. This kind of interaction is known as the dihydrogen bond [80–83] and really, early on the LiH…H–H system was classified as linked by this type of interaction [84].

Fig. 15.22

The molecular graph of the LiH…H₂ complex, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the isolines of Laplacian of electron density are also presented

The distribution of the laplacian of electron density (Fig. 15.22) shows the vertical to the H–H σ-bond concentration of the electron density and this density depletion at the edges of H-atoms of hydrogen molecule. There is a very interesting distribution for the LiH species where it is clear that hydride anion interacts with the electron density deprived Li cation. The most important orbital-orbital interaction corresponds here to the σ_Li–H → σ ^*_H–H overlap where σ_Li–H is the σ-bond orbital of LiH molecule while σ ^*_H–H is the antibonding H–H orbital of dihydrogen, the corresponding energy of interaction is equal to 1.7 kcal/mol. The polarization of the H₂ molecule is observed here since the NBO atomic charges are equal to +0.031 au and −0.036 au (the positively charged H-atom is in contact with the negatively charged H-atom of the LiH species). One can see that the whole H₂ molecule is slightly negatively charged, this is because of the acidic properties of H₂ in this complex what results in the electron density shift from the LiH species to the dihydrogen. This is worth mentioning that such polarization does not occur for the dihydrogen in the NH₄ ⁺…H₂ complex considered earlier here since atomic charges of H-atoms in the dihydrogen are both equal to +0.007 au. In the latter complex the H₂ molecule is slightly positively charged since there is outflow of the electron density from that species due to its Lewis base characteristics.

The analysis of interactions of dihydrogen with other species seems to be very important to understand the mechanisms of numerous reactions and processes such as for example, the activation of molecular hydrogen and next its splitting at reaction center [85]; usually the metallic center is considered both experimentally [86] and theoretically [87].

Figure 15.23 presents molecular graphs of two complexes where dihydrogen interacts with the phosphorus (non-metallic) center, i.e. the PFH₂…H₂ and PF(CH₃)₂…H₂ complexes are considered. In both cases the P-center is characterized by the Lewis acid properties due to the existence of the σ-hole since the regions of the positive EP situated in the elongation of F–P bond are observed. The σ-holes located at the P-centers are characterized by the positive EPs which amount 0.060 au and 0.037 au for the PFH₂ and PF(CH₃)₂ monomers, respectively. The binding energies for the PFH₂–H₂ and PF(CH₃)₂–H₂ systems are equal to −1.2 kcal/mol and −0.7 kcal/mol, respectively (not corrected for BSSE).

Fig. 15.23

The molecular graphs of the PFH₂…H₂ (left) and PF(CH₃)₂…H₂ (right) complexes, solid and broken lines correspond to bond paths, big circles to attractors and small green circles to BCPs, the reactive surface (∇ ²ρ(r) = 0 isosurface) is presented

There is an interesting issue if the dihydrogen in those complexes reveals its Lewis acid or Lewis base properties. Figure 15.23 shows that for the PFH₂–H₂ complex the P-dihydrogen intermolecular bond path crosses the region of the σ-hole but very close to the region of the electron charge concentration corresponding to the unshared P electron pair and next this bond path is aimed to the BCP of dihydrogen; however finally it ends at the H-atom attractor. The non-linear trajectory of the bond path does not show ambiguously the nature of centers being in contact; one may rather expect that the dihydrogen interacts as the Lewis base with the Lewis acid region of phosphorus. Two most important orbital-orbital interactions are detected for this complex; σ(H₂) → σ^*(P–F) and n(P) → σ^*(H₂) which correspond to the Lewis base and Lewis acid properties of dihydrogen, respectively. The energy of interaction for the former overlap is equal to 1.5 kcal/mol while it amounts 0.6 kcal/mol for the latter overlap. Thus according to the NBO approach, the dihydrogen reveals stronger Lewis base properties than the Lewis acid ones in the PFH₂…H₂ complex.

The different picture is observed for the PF(CH₃)₂–H₂ complex where the trajectory of the P…dihydrogen bond path prompts the Lewis acid properties of H₂ and Lewis base properties of the phosphorus species since the bond path crosses the lone electron pair region of P-center and it clearly ends at one of H-atoms of the dihydrogen. The NBO results confirm that since the σ(H₂) → σ^*(P–F) and n(P) → σ^*(H₂) orbital-orbital energies of interactions are equal to 0.2 kcal/mol and 0.6 kcal/mol, respectively showing the stronger Lewis acid properties of the dihydrogen in that complex.

The interactions of molecular hydrogen with the PF(CH₃)₂ and PFH₂ species are classified as weak ones thus there are only slight electron density shifts for the corresponding complexes, -0.002 au and +0.008 au for the PFH₂–H₂ and PF(CH₃)₂–H₂ complexes, respectively (NBO results). This means that for the PFH₂–H₂ complex there is the shift from the H₂ molecule while for the PF(CH₃)₂–H₂ complex the shift to the dihydrogen is observed. Those shifts clearly confirm the results presented earlier here of the Lewis base or Lewis acid properties of the molecular hydrogen.

5 Summary

The analysis of the location of bond paths, of their trajectories and the distribution of the electron density in species analyzed, especially if such a distribution is discussed on the basis of the laplacian of the electron density, are very useful tools to deepen the understanding of the nature of interactions. This is also important that the analysis of the molecular graphs is often in line with the results of other approaches. This is worth mentioning that the molecular graph may be treated as a source of information on the location of atomic attractors (positions of atoms) and on the location of bond paths corresponding to the preferable stabilizing interactions.

Let us look at examples presented in this chapter; it is evident, that very often for weak interactions the atom-atom contacts correspond to Lewis acid–Lewis base centers’ contacts characterized by the opposite electrostatic potentials. The hydrogen bonded systems are examples of such a situation, or even more generally, the σ-hole bonds. This is interesting that the bond paths, for weak interactions, usually connect the local maximum of EP with the minimum of EP. The complexes of the BrF₅ characterized by the pentavalent bromine center are examples of that situation; there are four equivalent local maxima of EP at the Br center situated symmetrically around the fourfold axis of symmetry of the BrF₅ molecule. This is why the complexes of BrF₅ with weak Lewis bases are not linear since the bond path crosses one of the local maxima of bromine. However the interactions with stronger Lewis bases like F⁻ or Cl⁻ anions lead to the meaningful changes in the electron charge distribution what results in the formation of BrF₆ ⁻ and BrF₅Cl⁻ complexes with the F–Br…F⁻ and F–Br…Cl⁻ linear arrangements, respectively.

One can enumerate other examples where the molecular graphs inform of the kinds and of the nature of interactions. The complexes of molecular hydrogen may be mentioned here. The dihydrogen possesses Lewis acid and Lewis base characteristics since the positive electrostatic potential is observed at the H-atoms, at edges of the H–H bond while the negative EP is connected with the σ-bond. This is why the H₂ molecule interacts with Lewis acid centers through its σ-electrons and in the corresponding molecular graphs the bond paths between the Lewis acid center and the bond critical point of the H–H σ-bond are observed. On the other hand if the H₂ molecule reveals their Lewis acid properties thus the bond paths between H-atom of molecular hydrogen and the Lewis base center may be observed.

In general the contour maps of the laplacian of the electron density nicely show the links between the regions of the electron density depletion with the regions of its concentration; this was rationalized recently by Eskandari and coworkers in terms of the lump-hole interactions concept [55, 56].