The Nature of Triel Bonds, a Case of B and Al Centres Bonded with Electron Rich Sites

The second-order Møller–Plesset perturbation theory calculations with the aug-cc-pVTZ basis set were performed on complexes of triel species: BCl3, BH3, AlCl3, and AlH3 acting as Lewis acids through the B or Al centre with Lewis base units: NCH, N2, NH3, and Cl− anion. These complexes are linked by triel bonds: B/Al⋅⋅⋅N or B/Al⋅⋅⋅Cl. The Quantum Theory of ´Atoms in Molecules´ approach, Natural Bond Orbital method, and the decomposition of energy of interaction were applied to characterise the latter links. The majority of complexes are connected through strong interactions possessing features of covalent bonds and characterised by short intermolecular distances, often below 2 Å. The BCl3⋅⋅⋅N2 complex is linked by a weak interaction corresponding to the B⋅⋅⋅N distance of ~3 Å. For the BCl3⋅⋅⋅NCH complex, two configurations corresponding to local energetic minima are observed, one characterised by a short B⋅⋅⋅N distance and a strong interaction and another one characterised by a longer B⋅⋅⋅N distance and a weak triel bond. The tetrahedral triel structure is observed for complexes linked by strong triel bonds, while, for complexes connected by weak interactions, the structure is close to the trigonal pyramid, particularly observed for the BCl3⋅⋅⋅N2 complex.


Introduction
The number of experimental and theoretical studies concerning inter-and intramolecular interactions increased rapidly in recent years [1][2][3][4][5][6][7]. The concept of σ-holes and π-holes is one of the most important trials to explain nature of the broad spectrum of interactions [2,3]. The aerogen (group 18), halogen (group 17), chalcogen (group 16), pnicogen (group 15), and tetrel (group 14) atoms [1][2][3][4][5][6][7] may act as the Lewis acid centres interacting with electron-rich sites; these interactions taking their names from names of corresponding centres are often classified as σ-hole bonds [2,3]. In the above-mentioned centres, the depletion of the electron charge may be observed on their edges in directions of their bonds with other atoms. These sites are just named as σ-holes; the above-described electron charge outflow is often sufficient to lead to the positive electrostatic potential (EP) at the σ-holes and, thus, to their Lewis acid properties.
It may be confusing that the centres described here play a role of Lewis acids since the majority of them are electronegative atoms. Some of these centres may be characterised by π-holes that are also regions of the depletion of the electron charge and, consequently, often of the positive electrostatic potential. The π-holes occur for planar species or planar molecular fragments in directions perpendicular to the corresponding planes [2,3]. The elements of the 13th group in their numerous compounds are characterised by the occurrence of π-holes. The corresponding interactions named as triel bonds are also classified as π-hole bonds [8][9][10]. The triel bonds usually designate interactions The calculations were performed for complexes of the BCl3, BH3, AlCl3, and AlH3 species acting as the Lewis acid units with the following Lewis bases: NCH, N2, NH3, and Cl − anion. Figure 1   . presents maps of the electrostatic potential (EP) for the molecular surfaces of Lewis acid units characterised by the electron density of 0.001 au. One can see regions of positive EP at the triel centres that correspond to the π-holes; values of the maximum EP are also presented. These positive EPs at π-holes determine the Lewis acid properties of triel centres that may be connected with electron-rich sites. The EP value is greater for trihydride than for the corresponding trichloride; moreover, EP is greater at the aluminium centre than at the boron centre of the corresponding species. The lowest maximum value of EP for the BCl3 species may be explained by the influence of chlorine substituents on the boron centre. However, there is no clear dependence of the EP value on the strength of the corresponding triel bond, as results discussed further here show. Thus, this means that not only electrostatic forces determine the strength of these interactions.
The energetic parameters of complexes linked by triel bonds and analysed here are presented in Table 1. The interaction and binding energies, Eint and Ebin [29], respectively, are included, as well as the deformation energy, Edef [30], and the basis set superposition error correction, BSSE [31]. The interaction energy is the difference between the energy of the complex and the sum of energies of interacting units with their geometries taken from the geometry of the complex. One may say that it describes the strength of interaction of the link in complex. In the binding energy, the energies of the interacting units correspond to their geometries optimised separately. In such a way, Ebin takes into account the deformation of these units resulting from the complexation, Ebin = Eint + Edef. The latter term, the deformation energy, may be understood as "the energetic cost" of the complex formation. There are systems where Eint is negative, indicating the local stabilising interaction, while Ebin is positive, showing that the formation of the complex is not energetically preferred [32]; this may correspond to an endothermic reaction. This is not a case for complexes analysed here, where both The calculations were performed for complexes of the BCl3, BH3, AlCl3, and AlH3 species acting as the Lewis acid units with the following Lewis bases: NCH, N2, NH3, and Cl − anion. Figure 1 and Scheme 1 present a few examples of systems analysed here.  . presents maps of the electrostatic potential (EP) for the molecular surfaces of Lewis acid units characterised by the electron density of 0.001 au. One can see regions of positive EP at the triel centres that correspond to the π-holes; values of the maximum EP are also presented. These positive EPs at π-holes determine the Lewis acid properties of triel centres that may be connected with electron-rich sites. The EP value is greater for trihydride than for the corresponding trichloride; moreover, EP is greater at the aluminium centre than at the boron centre of the corresponding species. The lowest maximum value of EP for the BCl3 species may be explained by the influence of chlorine substituents on the boron centre. However, there is no clear dependence of the EP value on the strength of the corresponding triel bond, as results discussed further here show. Thus, this means that not only electrostatic forces determine the strength of these interactions.
The energetic parameters of complexes linked by triel bonds and analysed here are presented in Table 1. The interaction and binding energies, Eint and Ebin [29], respectively, are included, as well as the deformation energy, Edef [30], and the basis set superposition error correction, BSSE [31]. The interaction energy is the difference between the energy of the complex and the sum of energies of interacting units with their geometries taken from the geometry of the complex. One may say that it describes the strength of interaction of the link in complex. In the binding energy, the energies of the interacting units correspond to their geometries optimised separately. In such a way, Ebin takes into account the deformation of these units resulting from the complexation, Ebin = Eint + Edef. The latter term, the deformation energy, may be understood as "the energetic cost" of the complex formation. There are systems where Eint is negative, indicating the local stabilising interaction, while Ebin is positive, showing that the formation of the complex is not energetically preferred [32]; this may correspond to an endothermic reaction. This is not a case for complexes analysed here, where both interaction and binding energies are negative for all cases (Table 1).  . presents maps of the electrostatic potential (EP) for the molecular surfaces of Lewis acid units characterised by the electron density of 0.001 au. One can see regions of positive EP at the triel centres that correspond to the π-holes; values of the maximum EP are also presented. These positive EPs at π-holes determine the Lewis acid properties of triel centres that may be connected with electron-rich sites. The EP value is greater for trihydride than for the corresponding trichloride; moreover, EP is greater at the aluminium centre than at the boron centre of the corresponding species. The lowest maximum value of EP for the BCl 3 species may be explained by the influence of chlorine substituents on the boron centre. However, there is no clear dependence of the EP value on the strength of the corresponding triel bond, as results discussed further here show. Thus, this means that not only electrostatic forces determine the strength of these interactions.  There is a sub-group of complexes analysed here, with anions being products of interactions between the trivalent triel species and the chloride anion. The large −Eint and −Ebin values indicate that they are stable systems with interactions corresponding to exothermic reactions. Large deformation energies are observed for these anionic systems. In the BCl4 − anion, this energy is close to 40 kcal/mol. One can see that the formation of these anions leads to great deformations of geometries of interacting units where the planar trichlorides and trihydrides are transformed into the tetrahedral structures. For example, the BCl4 − (see Scheme 1) and AlCl4 − anions are characterised by the Cl-B/Al-Cl angle of 109.5° as for methane and other ideal tetrahedral species. The corresponding H-B/Al-Cl angle for the BH3Cl − and AlH3Cl − anions is equal to 106.5° and 107.1°, respectively ( Table 2). These anions, being The energetic parameters of complexes linked by triel bonds and analysed here are presented in Table 1. The interaction and binding energies, E int and E bin [29], respectively, are included, as well as the deformation energy, E def [30], and the basis set superposition error correction, BSSE [31]. The interaction energy is the difference between the energy of the complex and the sum of energies of interacting units with their geometries taken from the geometry of the complex. One may say that it describes the strength of interaction of the link in complex. In the binding energy, the energies of the interacting units correspond to their geometries optimised separately. In such a way, E bin takes into account the deformation of these units resulting from the complexation, E bin = E int + E def . The latter term, the deformation energy, may be understood as "the energetic cost" of the complex formation. There are systems where E int is negative, indicating the local stabilising interaction, while E bin is positive, showing that the formation of the complex is not energetically preferred [32]; this may correspond to an endothermic reaction. This is not a case for complexes analysed here, where both interaction and binding energies are negative for all cases (Table 1).
There is a sub-group of complexes analysed here, with anions being products of interactions between the trivalent triel species and the chloride anion. The large −E int and −E bin values indicate that they are stable systems with interactions corresponding to exothermic reactions. Large deformation energies are observed for these anionic systems. In the BCl 4 − anion, this energy is close to 40 kcal/mol.
One can see that the formation of these anions leads to great deformations of geometries of interacting units where the planar trichlorides and trihydrides are transformed into the tetrahedral structures.  (Table 2). These anions, being the result of interaction of trivalent B and Al species with chloride, are stable tetravalent systems where the triel centre obeys the octet rule. Table 1. The energetic characteristics of complexes analysed (in kcal/mol), the interaction and binding energy (E int and E bin , respectively), the deformation energy (E def ), and the basis set superposition error (BSSE); a and b superscripts stand for complexes linked by weak and strong triel bonds, respectively. Table 2. The geometrical and the Natural Bond Orbital (NBO) parameters; B/Al···N/Cl is the distance between the triel centre and the centre of the Lewis base unit (in Å), α-angle (in degrees) is defined in Scheme 1, Ch-shift is the electron charge shift from the Lewis base unit to the Lewis acid that results from complexation (au), while B/Al is the charge (in au) of the triel centre in the complex, 1 and Pol-B/Al is the percentage of the electron density of the B/Al···N/Cl bond orbital at the triel centre; a and b superscripts stand for complexes linked by weak and strong triel bonds, respectively.  Table 1 shows that other, neutral complexes are characterised by a broad spectrum of interactions, from −E int amounting to 1.5 kcal/mol for the BCl 3 ···N 2 complex, to 50.4 kcal/mol for the BCl 3 ···NH 3 one. The deformation energy for neutral complexes is greater for stronger interactions. For the BCl 3 ···NH 3 complex, the E def value amounts 23.6 kcal/mol. For the very weak interaction in the BCl 3 ···N 2 complex, E def is negligible, equal to 0.02 kcal/mol, while, for another weak interaction in the AlH 3 ···N 2 complex, it amounts to 0.7 kcal/mol. The complexes linked by weak interactions are marked in all tables by superscript a, while those classified as connected by strong interactions are marked by superscript b. However, there is no sharp border between strong and weak interactions, and the division proposed here is rather contractual. These relationships between the deformation energy and the parameters describing strength of interaction were observed earlier for complexes linked by triel bonds [10,24]. The BSSE correction is located in the range 1-2 kcal/mol for the majority of systems considered here; for a few boron complexes, it is greater than 2 kcal/mol, while, for a few aluminium species, it is lower than 1 kcal/mol.

Complex B/Al···N/Cl
There are two configurations of the BCl 3 ···NCH complex, characterised by the energies of interaction amounting to −23.6 kcal/mol and −3.5 kcal/mol with the corresponding B···N distances of 1.628 Å and 2.817 Å, respectively (Table 2). However, the binding energies of these two configurations are almost equal since, for that characterised by the shorter B···N distance, the complexation is connected with great changes of geometries of interacting units. The Cl-B-N angle amounts here to 104 • , and the deformation energy is equal to 19.8 kcal/mol. For the configuration with the longer B···N distance, these values are equal to 91.7 • and 0.3 kcal/mol, respectively.
The angle mentioned above is designated as α hereinafter; it is between the Cl/H-B/Al bond of the Lewis acid unit and the B/Al···N/Cl intermolecular distance (Scheme 1). In the case of strong interactions, complexes are close to the tetrahedron structure, while, in the case of weak interactions, the Lewis acid unit remains planar, or nearly so, and the α angle defined above is equal or close to 90 • . For two configurations of the BCl 3 ···NCH complex, one of them is close to the tetrahedral structure and the other one is close to the trigonal pyramid with the Lewis acid unit close to the planarity.
The α angles are collected in Table 2, while the intermolecular distances are also included there, as well as the selected the Natural Bond Orbital (NBO) [33,34] parameters. If one refers to the consistent van der Waals radii proposed by Truhlar and co-workers (N-1.55 Å, Cl-1.75 Å, B-1.92 Å, Al-1.84 Å) [35], all intermolecular distances are shorter than the corresponding sum of van der Waals radii. However, the term "intermolecular" should be used with caution since, for the tetrahedral anionic species, all links to the triel centre possess characteristics of covalent bonds. The stability of the similar anion, BF 4 − , was discussed recently [36]. species. It was pointed out that trigonal boron trihalides occur only in the gas phase [37]; it seems they are very reactive possessing strong Lewis acid properties, and they react with nucleophiles in more condensed phases. The electron charge shift from the Lewis base to the Lewis acid unit that results from complexation calculated within the NBO approach is included in Table 2. One can see that greater electron charge shifts are observed for stronger interactions. The greatest shifts between −0.38 au and −0.69 au occur for anionic complexes, as well as for the BCl 3 ···NH 3 and BH 3 ···NH 3 complexes, while the smallest ones, lower than −0.01 au, occur for complexes linked by the weakest interactions, such as the BCl 3 ···N 2 complex and the BCl 3 ···NCH configuration characterised by the longer B···N distance. Table 2 presents the atomic charge of the Lewis acid centre, boron, and aluminium. For all complexes analysed, their formation leads to this charge decrease, up to negative values for the boron centre in the BH 3 complexes. There are only two exceptions to the above rule; for the weakest interactions in the BCl 3 ···N 2 complex and in one of the BCl 3 ···NCH configurations, the complexation leads to the increase of the positive charge of the boron centre. The Lewis acid properties of the triel centres are supported by the positive electrostatic potentials (EPs) at their surfaces that are related to the above-mentioned π-holes; the positive EPs for simple species of boron and other triel elements were analysed in former studies in detail [8][9][10]38].
It is worth recalling that, for the latter two cases of weakest interactions, the smallest α-angles occur; thus, the BCl 3 part of the complex is close to the planarity. For all remaining complexes, the NBO approach detects the σ-bond orbital between the boron or aluminium centre and the nitrogen or chlorine Lewis base site. The polarisation of the latter link is shown in Table 2. It is understood as the percentage of the electron density at the triel (B or Al) centre. One can see that there is no clear tendency here. The greatest concentrations of the electron density at the triel centre occur for the AlCl 3 complexes, as well as for the AlH 3 ···N 2 and AlH 3 ···NCH complexes, above 90% (except of the AlCl 4 − complex where it amounts to 83.5%). In the case of AlH 3 ···NH 3 and AlH 3 Cl − , the polarisation is equal to 8.5% and 15.1%, respectively, which means that the electron density for the Al-N and Al-Cl links, respectively, is accumulated mainly at nitrogen and chlorine centres. The analysis of these intermolecular links that may be classified as triel bonds, based on results collected in Tables 1 and 2, shows that their properties for aluminium and boron centres differ.

Double Minima for Triel-Lewis Base Potential Energy Curves
It was mentioned earlier here that there are examples of complexes of simple boron species that are characterised by two configurations corresponding to energetic minima, i.e., by the double minimum of potential energy [19][20][21][22]38]. This also concerns simple moieties of elements of the fourth and 14th groups [23]. In the case of triel species, two configurations were discussed for the BCl 3 ···CH 3 CN and BF 3 ···CH 3 CN complexes; however, in the latter case, one of the potential energy minima is not well separated [20][21][22]. Two configurations were shortly discussed recently for the BCl 3 ···NCH complex, and two local energy minima were also found for the BBr 3 ···NCH and BI 3 ···NCH complexes [38]. It was concluded that the existence of two configurations in simple triel species is the result of the balance of interaction energy terms, and that Pauli repulsion plays a crucial role [19]. In a more recent study, it was discussed that the double minimum results from the balance of electrostatic interactions between the BX 3 (X = Cl, Br, I) and HCN units [38].
Potential energy curves were constructed for all complexes analysed here, and the double minimum was found only in one case, i.e., for the BCl 3 ···NCH complex. This curve and other potential energy curves for the three additional complexes are presented in Figure 3. Because of differences in the scale for each complex considered, the curves for BH 3 ···NCH, AlCl 3 ···NCH, and AlH 3 ···NCH complexes are restricted only to the proximity of energetic minima. Each of the curve is "normalised" separately. This means that, for each complex, the differences between their energies and the energetic minimum are plotted versus the corresponding Al/B···N distances. BCl3⋅⋅⋅NCH complex, and two local energy minima were also found for the BBr3⋅⋅⋅NCH and BI3⋅⋅⋅NCH complexes [38]. It was concluded that the existence of two configurations in simple triel species is the result of the balance of interaction energy terms, and that Pauli repulsion plays a crucial role [19]. In a more recent study, it was discussed that the double minimum results from the balance of electrostatic interactions between the BX3 (X = Cl, Br, I) and HCN units [38]. Potential energy curves were constructed for all complexes analysed here, and the double minimum was found only in one case, i.e., for the BCl3⋅⋅⋅NCH complex. This curve and other potential energy curves for the three additional complexes are presented in Figure 3. Because of differences in the scale for each complex considered, the curves for BH3⋅⋅⋅NCH, AlCl3⋅⋅⋅NCH, and AlH3⋅⋅⋅NCH complexes are restricted only to the proximity of energetic minima. Each of the curve is "normalised" separately. This means that, for each complex, the differences between their energies and the energetic minimum are plotted versus the corresponding Al/B⋅⋅⋅N distances. It seems that the existence of two potential energy minima for the BCl3⋅⋅⋅NCH complex results from the influence of chlorine substituents that obscure the boron centre. One may consider that the interaction between the BCl3 and NCH units contains the electrostatic interactions between the nitrogen centre with the boron centre and with the chlorine substituents. The first N⋅⋅⋅B interaction is attractive, while the N⋅⋅⋅Cl interactions are repulsive. This is the electrostatic balance mentioned above here. In contrast, the chlorine substituents cannot obscure the aluminium centre that is characterised by a greater volume than the boron one. However, it seems that the transfer from one configuration characterised by the shallow minimum into the configuration corresponding to the deeper one in the BCl3⋅⋅⋅NCH complex may be easily forced, for example, by crystal structure forces,   It seems that the existence of two potential energy minima for the BCl 3 ···NCH complex results from the influence of chlorine substituents that obscure the boron centre. One may consider that the interaction between the BCl 3 and NCH units contains the electrostatic interactions between the nitrogen centre with the boron centre and with the chlorine substituents. The first N···B interaction is attractive, while the N···Cl interactions are repulsive. This is the electrostatic balance mentioned above here. In contrast, the chlorine substituents cannot obscure the aluminium centre that is characterised by a greater volume than the boron one. However, it seems that the transfer from one configuration characterised by the shallow minimum into the configuration corresponding to the deeper one in the BCl 3 ···NCH complex may be easily forced, for example, by crystal structure forces, since the potential barrier height for such a transfer amounts to 1.6 kcal/mol only. Figure 4 presents the potential energy curve only for the BCl 3 ···NCH complex. However, the potential energy curve in Figure 4 is more flattened than the corresponding one in Figure 3 because of scale reasons. Another curve presented in Figure 4 also presents energies of this complex for various B···N distances; however, the rigid HCN and BCl 3 moieties are considered as possessing geometries corresponding to the isolated species in their energetic minima. This means that these geometries do not change for different B···N distances. This curve is normalised according to the global energy minimum of the complex. One can recall that, in the case of the curve with two minima for each B···N distance, the other geometrical parameters are relaxed (optimised). One can see (Figure 4) that, for the rigid interacting units, a monotonic increase of energy is observed with the shortening of the B···N distance. This means that the increase of repulsion between chlorine centres and nitrogen is not compensated for by the boron···nitrogen attraction. In the case of "relaxed" geometries, the α-angle increases with the shortening of the B···N distance. The latter results in weaker Cl···N repulsions than for "the rigid case". These repulsions for relaxed geometries are compensated for by the B···N attraction, resulting in the occurrence of the second deeper potential energy well for the B···N distance equal to 1.628 Å (see Table 2). However, other interaction energy terms should also be taken into account. The Pauli repulsion is weaker for relaxed geometries than for the rigid interacting species, whereas the orbital energy related to the electron charge shifts is also more important for shorter B···N distances.  Table 2). However, other interaction energy terms should also be taken into account. The Pauli repulsion is weaker for relaxed geometries than for the rigid interacting species, whereas the orbital energy related to the electron charge shifts is also more important for shorter B⋅⋅⋅N distances.

Covalency of Triel Bonds
The former results discussed here show that the majority of triel bonds possess characteristics of covalent bonds. This is in agreement with the experimental data since the X-ray and neutron diffraction results concerning crystal structures indicate that coordination four occurs most often for the triel centres [24]; in other words, the trivalent triel centres are very reactive and they interact strongly with nucleophiles.
Quantum Theory of Atoms in Molecules (QTAIM) [39,40] calculations were performed here at the level corresponding to the optimised geometries, MP2/aug-cc-pVTZ -the second-order Møller-Plesset perturbation theory (MP2) calculations with the aug-cc-pVTZ basis set. Table 3 shows the characteristics of Al/B⋅⋅⋅N/Cl bond critical points (BCPs) for links between the trivalent triel units and the Lewis base species. The electron density at the corresponding BCP, ρBCP, is situated between 0.006 au for the BCl3⋅⋅⋅N2 complex and 0.131 au for the BCl3⋅⋅⋅NH3 complex. The first value is typical for weak van der Waals interactions, while the latter one is typical for covalent bonds that are characterised by values of ∼0.1 au [39,40]. Few complexes analysed in this study possess ρBCP values

Covalency of Triel Bonds
The former results discussed here show that the majority of triel bonds possess characteristics of covalent bonds. This is in agreement with the experimental data since the X-ray and neutron diffraction results concerning crystal structures indicate that coordination four occurs most often for the triel centres [24]; in other words, the trivalent triel centres are very reactive and they interact strongly with nucleophiles.
Quantum Theory of Atoms in Molecules (QTAIM) [39,40] calculations were performed here at the level corresponding to the optimised geometries, MP2/aug-cc-pVTZ -the second-order Møller-Plesset perturbation theory (MP2) calculations with the aug-cc-pVTZ basis set. Table 3 shows the characteristics of Al/B···N/Cl bond critical points (BCPs) for links between the trivalent triel units and the Lewis base species. The electron density at the corresponding BCP, ρ BCP , is situated between 0.006 au for the BCl 3 ···N 2 complex and 0.131 au for the BCl 3 ···NH 3 complex. The first value is typical for weak van der Waals interactions, while the latter one is typical for covalent bonds that are characterised by values of 0.1 au [39,40]. Few complexes analysed in this study possess ρ BCP values outweighing 0.1 au. The typical QTAIM characteristic of the covalent character of the atom-atom link is the negative value of the Laplacian of electron density at the corresponding BCP, ∇ 2 ρ BCP . This negative value that indicates the concentration of the electron density in the interatomic region is observed only for the BCl 4 − anion. However, it is often assumed that, even for positive ∇ 2 ρ BCP values, the negative value of the total electron energy density at the BCP, H BCP , confirms the partly covalent character of interaction [41][42][43]. The majority of the links presented in Table 3 are characterised by the negative H BCP values. The positive H BCP values correspond to the weakest interactions. However, there is no correlation between binding or interaction energy and the QTAIM characteristics: ρ BCP , ∇ 2 ρ BCP , or H BCP . For example, the linear correlation coefficient for the dependence between the ρ BCP and interaction energy, E int , is equal to 0.517. The ρ BCP parameter is related to the part of the interaction connected with its covalent character and not to the total interaction energy [44]. It seems that, for the part of complexes analysed here, the electrostatic interaction is more important than the term related to covalency-the orbital energy. For example, for the BCl 3 ···NH 3 complex mentioned above, the ρ BCP value amounts to 0.131 au, while, for the AlCl 3 ···NH 3 complex, it is equal to 0.060 au ( Table 3). The interaction and binding energies for these systems are very close to each other (Table 1). However, the electron density at the BCP concerns the covalent character of interaction that is greater for the boron system than for the aluminium one. The α-parameter presented earlier here (Scheme 1) also expresses the covalent character of the interaction since it informs about the transformation from the triel planar trivalent system into the tetrahedral structure. Figure 5 shows the excellent exponential correlation between α and ρ BCP . The anionic systems significantly differing from the remaining complexes are excluded from this dependence; they are presented in this figure only for comparison. The results of the decomposition of interaction energies are shown in Table 4. The decomposition was performed for the density functional theory (DFT) calculations. However, there is an excellent agreement between the DFT and MP2/aug-cc-pVTZ results; for example, a linear correlation between DFT and MP2 interaction energies is observed, R 2 = 0.981. Thus, the results of decomposition are well fitted to the MP2 results of calculations discussed here.
interaction since it informs about the transformation from the triel planar trivalent system into the tetrahedral structure. Figure 5 shows the excellent exponential correlation between α and ρBCP. The anionic systems significantly differing from the remaining complexes are excluded from this dependence; they are presented in this figure only for comparison. The results of the decomposition of interaction energies are shown in Table 4. The decomposition was performed for the density functional theory (DFT) calculations. However, there is an excellent agreement between the DFT and MP2/aug-cc-pVTZ results; for example, a linear correlation between DFT and MP2 interaction energies is observed, R 2 = 0.981. Thus, the results of decomposition are well fitted to the MP2 results of calculations discussed here.  Let us return to the BCl 3 ···NH 3 and AlCl 3 ···NH 3 pair of complexes discussed above here. They are characterised by a similar strength of interaction; however, for the former one, the ρ BCP is over two times greater than for the latter complex. A negative H BCP is observed for the former complex and it is also negative for the latter one, but the latter value is very close to zero. This may suggest that, for the boron moiety, the B···N contact is covalent in nature, at least partly, while the Al···N link for the latter complex is electrostatic in nature. This also supports the suggestion of Gillespie and Popelier [44] that electron density at the bond critical point expresses the covalent nature of interaction. Furthermore, for the aluminium complex, the electrostatic interaction energy term is almost two times greater than the orbital energy, which concerns the absolute values (Table 4). In the case of the BCl 3 ···NH 3 complex, another situation is observed, where orbital and electrostatic energies are comparable to each other. The results of Table 4 show that there is no specific difference between boron and aluminium complexes or between the triel centres connected with chlorine or hydrogen substituents. One can also see that the dispersion energy is much less important with respect to stabilising the systems considered than the electrostatic and orbital terms; this is observed for all complexes analysed. Figure 6 presents linear correlations between the electron density at the B/Al···N/Cl BCP, ρ BCP , and two terms of the interaction energy resulting from its partitioning: Pauli repulsion (∆E Pauli ) and orbital energy (∆E orb ) (see Computational Details section). These are good correlations despite them concerning the sample of complexes containing various kinds of interactions. These two terms, ∆E Pauli and ∆E orb , are often related to the covalent character of interaction [43], especially the orbital energy, which is connected with the electron charge shifts resulting from complexation. It was pointed out in several studies that the dominance of the ∆E orb term testifies to the covalent character of interaction [43]. It is worth mentioning that ρ BCP does not correlate with ∆E elstat , with the ∆E disp term, or with the total interaction energy, as mentioned above.
interaction. Furthermore, for the aluminium complex, the electrostatic interaction energy term is almost two times greater than the orbital energy, which concerns the absolute values (Table 4). In the case of the BCl3⋅⋅⋅NH3 complex, another situation is observed, where orbital and electrostatic energies are comparable to each other. The results of Table 4 show that there is no specific difference between boron and aluminium complexes or between the triel centres connected with chlorine or hydrogen substituents. One can also see that the dispersion energy is much less important with respect to stabilising the systems considered than the electrostatic and orbital terms; this is observed for all complexes analysed. Figure 6 presents linear correlations between the electron density at the B/Al⋅⋅⋅N/Cl BCP, ρBCP, and two terms of the interaction energy resulting from its partitioning: Pauli repulsion (ΔEPauli) and orbital energy (ΔEorb) (see Computational Details section). These are good correlations despite them concerning the sample of complexes containing various kinds of interactions. These two terms, ΔEPauli and ΔEorb, are often related to the covalent character of interaction [43], especially the orbital energy, which is connected with the electron charge shifts resulting from complexation. It was pointed out in several studies that the dominance of the ΔEorb term testifies to the covalent character of interaction [43]. It is worth mentioning that ρBCP does not correlate with ΔEelstat, with the ΔEdisp term, or with the total interaction energy, as mentioned above.

Conclusions
The simple complexes of boron and aluminium trichlorides and trihydrides with Lewis base units such as hydrogen cyanide, molecular nitrogen, ammonia, and chloride anion were analysed. These complexes are linked through interactions that were named in recent studies as triel bonds. The majority of interactions analysed here are classified as strong ones that possess characteristics of covalent bonds. For the corresponding complexes of the latter interactions, the complexation leads to the transformation of the planar trigonal trichlorides and trihydrides into structures containing a tetrahedral triel centre (boron or aluminium). In particular, such a situation occurs for complexes of the chloride anion, while perfect tetrahedron structures are observed for BCl 4 − and AlCl 4 − species.
In a few cases, weak triel bonds are observed, especially for the BCl 3 ···N 2 complex and one of configurations of the BCl 3 ···NCH complex. The interactions linking the AlCl 3 ···N 2 and AlH 3 ···N 2 complexes may also be classified as weak ones. For these weak interactions, the Lewis acid unit, BCl 3 , AlCl 3 , or AlH3, retains almost a flat structure in the complex.
Different parameters may describe the character of triel bonds. Among numerous findings presented in this study, it is worth recalling that the angle parameter, α, describes the transformation from the trigonal planar structure into the tetrahedron; this parameter corresponds to the strength of interaction. It is discussed here that the electron density at the BCP corresponds rather to the part of the energy of interaction that is related to the electron charge shifts, and not to the total interaction energy.

Computational Details
The calculations were carried out with the use of Gaussian16 set of codes [45]. These are the second-order Møller-Plesset perturbation theory (MP2) [46] calculations with the aug-cc-pVTZ basis set [47]. Frequency calculations were performed at the same MP2/aug-cc-pVTZ level to confirm that the optimised structures of complexes correspond to the energetic minima.
The Quantum Theory of Atoms in Molecules (QTAIM) [39,40] was also used to analyse bond critical points (BCPs) of the intermolecular B/Al···N and B/Al···Cl contacts. The AIMAll program [48] was applied to carry out QTAIM calculations, as well as to calculate the electrostatic potentials (EPs). The analysis of the electron charge density shifts, being the result of complexation, was performed with the use of the Natural Bond Orbital (NBO) method [33,34]. The NBO calculations were performed at the BP86-D3/TZ2P level; i.e., with the use of the BP86 functional [49,50] in conjunction with the Grimme dispersion corrections (BP86-D3) [51] and the uncontracted Slater-type orbitals (STOs) as basis functions with triple-ζ quality for all elements [52]. The same BP86-D3/TZ2P level was applied to perform decomposition energy calculations. The NBO [53] and decomposition energy [54,55] calculations were carried out with the use of the ADF2017 program package [55,56] and using geometries of complexes optimised previously at the MP2/aug-cc-pVTZ level. The total interaction energy for the ADF partitioning is composed according to equation given below.
The term ∆E elstat corresponds to the quasi-classical electrostatic interaction between the unperturbed charge distributions of atoms; it is usually attractive (negative). The Pauli repulsion, ∆E Pauli , is the energy change associated with the transformation from the superposition of the unperturbed electron densities of the isolated fragments to the wave function that properly obeys the Pauli principle through antisymmetrisation and renormalisation of the product wave function. This repulsive term (positive) comprises the destabilising interactions between electrons of the same spin on either fragment. The orbital interaction, ∆E orb , corresponds to the charge transfer and polarisation effects.