Nature and Strength of Lewis Acid/Base Interaction in Boron and Nitrogen Trihalides

Abstract We have quantum chemically investigated the bonding between archetypical Lewis acids and bases. Our state‐of‐the‐art computations on the X3B−NY3 Lewis pairs have revealed the origin behind the systematic increase in B−N bond strength as X and Y are varied from F to Cl, Br, I, H. For H3B−NY3, the bonding trend is driven by the commonly accepted mechanism of donor−acceptor [HOMO(base)−LUMO(acid)] interaction. Interestingly, for X3B−NH3, the bonding mechanism is determined by the energy required to deform the BX3 to the pyramidal geometry it adopts in the adduct. Thus, Lewis acids that can more easily pyramidalize form stronger bonds with Lewis bases. The decrease in the strain energy of pyramidalization on going from BF3 to BI3 is directly caused by the weakening of the B−X bond strength, which stems primarily from the bonding in the plane of the molecule (σ‐like) and not in the π system, at variance with the currently accepted mechanism.


Introduction
The chemistry of Lewis acids and bases is rich and can be found in any general chemistry textbook. [1] In his epochal work, [2] Gilbert N. Lewis introduced the concept of electron-pair donoracceptor complexes, on which the current understanding of Lewis acid/base interactions is based. It defines Lewis acids as chemical species that accepts an electron-pair from a Lewis base to form a Lewis adduct. Thus, the Lewis acidity and basicity scales are associated with the stability of the adducts, that is, relative to a reference, a stronger Lewis acid or Lewis base forms a stronger bounded Lewis complex. The Lewis acid/ base chemistry has experienced continuous development since then [3] and has found utility in a wide range of research areas, including catalysis [4] and the recent advent of frustrated Lewis pair chemistry, [5] to name a few.
Due to the ubiquity of Lewis acid/base in chemistry, attempts to rationalize the nature and strength of this interaction abound. [6] The theory of hard and soft acids and bases (HSAB) proposed by Pearson [7] is undoubtedly the most popular qualitative model used to understand this interaction. The HSAB principle uses the intrinsic properties of the interacting species to explain the stability of acid/base complexes, namely, the concept of hardness and softness, which is based on properties such as size, polarizability, and electronegativity. In this model, a hard base (the term "hard" stands for small sized atoms with low polarizability and high electronegativity) would preferentially bind to a hard acid, while a soft base (the term "soft" stands for large sized atoms with high polarizability and low electronegativity) prefers to associate with a soft acid. However, the validity of this model has been questioned, as it has been shown to fail in predicting reactivity of archetypal reactions. [8] Interestingly, the relative Lewis acidity of boron trihalides with respect to strong bases (e. g., NH 3 , NMe 3 ) is known to increase along the series BF 3 < BCl 3 < BBr 3 ; however, the opposite trend is observed for the interaction with weak bases (e. g., N 2 , CH 3 F). [6i,9] This indicates that Lewis acid/base is a rather complex interaction that depends on the entire system, not only on the characteristics of the isolated acids and bases. Over the years, various theories have been proposed to explain the trends in stability of Lewis pairs involving boron trihalides, such as those based on π-backdonation, [9c,10] the ability to engage in stabilizing orbital interactions [11] or electrostatics, [9a] ligand close packing (LCP) model, [12] or electrophilicity principle. [13] The decreased Lewis acidity of BF 3 towards strong bases, compared to heavier boron trihalides, is widely attributed to a more efficient π charge donation from the fluorine lone-pair into the empty p orbital of the boron (π-backdonation), which reduces the availability of the boron atom to accept an electron pair from the Lewis base. [9c,10] However, it has been shown that the p (π)-p(π) overlap integral and the p(π) population at the boron is actually smaller for BF 3 than for BCl 3 . [11c,14] Alternatively, an intuitive argument based on the strength of frontier molecular orbital interactions has been proposed by Bessac and Frenking,[11b] that is, the energy of the LUMO of BX 3 decreases from X = F to Cl and results in more stabilizing orbital interactions with the HOMO of the Lewis base for BCl 3 compared to BF 3 . We note that these explanations are universal and neither can explain the reversal in Lewis acidities that is observed for the Lewis complexes between boron trihalides and weak bases.
We aim to illuminate the nature and strength of Lewis acid/ base interaction within the conceptual framework provided by KohnÀ Sham molecular orbital (KS-MO) theory and ultimately provide a unified framework to understand Lewis pairs. To this end, we investigate the underlying physical mechanism behind the formation of a systematic set of X 3 BÀ NY 3 Lewis pairs (Scheme 1, where X,Y = H, F, Cl, Br, and I). We first explore the archetypical boraneÀ ammonia adduct, H 3 BÀ NH 3 , and then separately evaluate the substituent effect on the Lewis acid and Lewis base by varying X,Y from H to F, Cl, Br, and I. To the best of our knowledge, this is the first thorough analysis on the formation of Lewis pairs involving the complete series of nitrogen and boron trihalides. Detailed analysis of the electronic structures and bonding mechanisms enable us to interpret our results in quantitative and chemically meaningful terms, which reveals the role of different components, namely, chargetransfer, electrostatic interaction and also strain energy, in the stability of the Lewis complexes. This demonstrates that, similar to hydrogen bonds, [15] Lewis acid/base interaction is a complex interplay of several energy components, whose importance depends on the molecular system and may not be easily captured in simple predictive models.

Computational Details
All calculations were performed using the Amsterdam Density Functional (ADF) software package. [16] Geometries and energies were calculated at the BLYP level of the generalized gradient approximation (GGA); exchange functional developed by Becke (B), and the GGA correlation functional developed by Lee, Yang and Parr (LYP). [17] The DFT-D3(BJ) method developed by Grimme and coworkers,[18] which contains the damping function proposed by Becke and Johnson, [19] was used to describe non-local dispersion interactions. Scalar relativistic effects are accounted for using the zeroth-order regular approximation (ZORA). [20] Molecular orbitals (MO) were expanded in a large uncontracted set of Slater type orbitals (STOs) containing diffuse functions: TZ2P. [21] The basis set is of triple-ξ quality for all atoms and has been augmented with two sets of polarization functions. All electrons were included in the variational process, i. e., no frozen core approximation was applied. An auxiliary set of s, p, d, f, and g STOs was used to fit the molecular density and to represent the Coulomb and exchange potentials accurately in each self-consistent field cycle. The accuracies of the fit scheme (ZLM fit) [22] and the integration grid (Becke grid) [23] were set to 'very good'. The Lewis acids were optimized with D 3h symmetry constraints, and the Lewis bases and Lewis adducts were optimized with C 3v symmetry constraints. All optimized structures were confirmed to be true minima through vibrational analyses [24] (no imaginary frequencies). The molecular structures were illustrated using CYLview. [25]

Activation Strain and Energy Decomposition Analysis
Insight into the nature of Lewis acid/base interaction is obtained by applying the activation strain model (ASM) [26] along the formation of the Lewis adducts. The formation of the Lewis pairs is computationally modelled by decreasing the distance between the boron atom of the Lewis acid and the nitrogen atom of the Lewis base, while other geometry parameters are included in the optimization. Thus, each analysis starts from an optimized Lewis acid and Lewis base at a relatively large distance, then, the BÀ N distance (r BÀ N ) is gradually decreased to a bond length smaller than the equilibrium distance of the Lewis adduct.
The activation strain model of chemical reactivity [26] is a fragmentbased approach to understand the energy profile of a chemical process in terms of the original reactants (i. e., the formation of the dimer from monomers). Thus, the overall bond energy ΔE(ξ) is decomposed into the respective total strain and interaction energy, ΔE strain (ξ) and ΔE int (ξ), and project these values onto the reaction coordinate ξ (in this case, r BÀ N ) [Eq. (1)].
In this equation, the total strain energy ΔE strain (ξ) is the penalty that needs to be paid to deform the reactants from their equilibrium structure to the geometry they adopt in the complex at point ξ of the reaction coordinate. On the other hand, the interaction energy ΔE int (ξ) accounts for all the chemical interactions that occur between the deformed fragments along the reaction coordinate.
The interaction energy between the deformed fragments is further analyzed in terms of quantitative KohnÀ Sham molecular orbital (KS-MO) theory in combination with a canonical energy decomposition analysis (EDA). [27] The EDA decomposes the ΔE int (ξ) into the following four physically meaningful energy terms [Eq. (3)]: Herein, ΔV elstat (ξ) is the classical electrostatic interaction between the unperturbed charge distributions of the (deformed) fragments and is usually attractive. The Pauli repulsion ΔE Pauli (ξ) comprises the destabilizing interaction between occupied closed-shell orbitals of both fragments due to the Pauli principle. The orbital interaction energy ΔE oi (ξ) accounts for polarization and charge transfer between the fragments, such as HOMOÀ LUMO interactions. It can be decomposed into the contributions from each irreducible Scheme 1. Formation of the Lewis pairs analyzed in this work.

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representation Γ of the interacting system [Eq. (4)]. Finally, the dispersion energy ΔE disp (ξ) accounts for the dispersion corrections as introduced by Grimme et al. [18] A detailed, step-by-step, guide on how to perform and interpret the ASM and EDA can be found in reference 26a. The Pyfrag program was used to facilitate these analyses. [28] DE oi ¼

Voronoi Deformation Density (VDD) charges
The atomic charge distribution was analyzed by using the Voronoi Deformation Density (VDD) method. [29] The VDD method partitions the space into so-called Voronoi cells, which are non-overlapping regions of space that are closer to nucleus A than to any other nucleus. The charge distribution is determined by taking a fictitious promolecule as reference point, in which the electron density is simply the superposition of the spherical atomic densities. The change in density in the Voronoi cell when going from this promolecule to the final molecular density of the interacting system is associated with the VDD atomic charge Q. Thus, the VDD atomic charge Q A VDD of atom A is given by: Instead of computing the amount of charge contained in an atomic volume, we compute the flow of charge from one atom to the other upon formation of the molecule. The physical interpretation is therefore straightforward. A positive atomic charge Q A corresponds to the loss of electrons, whereas a negative atomic charge Q A is associated with the gain of electrons in the Voronoi cell of atom A.

Structures and bond strengths
In this section, the geometries and bond energies of the X 3 BÀ NY 3 Lewis pairs (X,Y = H, F, Cl, Br, and I) are discussed. The results are summarized in Figure 1 (full structural data is provided in Table S1 in the Supporting Information). As BX 3 and NY 3 approach each other to form the Lewis adduct, the Lewis acid must pyramidalize from its trigonal planar equilibrium geometry, that is, the θ XÀ BÀ X angle decreases and the r BÀ X bond length increases (Table S1). This effect is much less pronounced in the Lewis base, as it already has a pyramidal equilibrium geometry and undergoes almost no deformation upon complexation. Our computed bond lengths and angles of boraneÀ ammonia (i. e., H 3 BÀ NH 3 ) are in very good agreement with existing experimental data [30]  The expected trends in Lewis adduct stabilities are nicely reproduced by our DFT computations at ZORA-BLYP-D3(BJ)/ TZ2P. BoraneÀ ammonia forms the strongest bond complex in our series of Lewis pairs (ΔE = À 29.5 kcal mol À 1 ). Upon substitution of the hydrogen atoms on the Lewis acid or Lewis base with halogen atoms, the energy of formation of the Lewis adduct ΔE decreases in strength, i. e., becomes less stabilizing, along the series: H, I, Br, Cl, F. The bond enthalpies at 298 K (ΔH 298.15 ) show the same trends as the electronic bond energies ΔE (see supporting methods and Table S1 in the Supporting Information). In the following sections, we partition the Lewis pairs into three sets: 1) H 3 BÀ NH 3 , 2) X 3 BÀ NH 3 , and 3) H 3 BÀ NY 3 (where X,Y = F, Cl, Br, and I), and provide a unified model to rationalize the strength of the Lewis pair bond through detailed analyses of the electronic structure and bonding mechanism.

Borane-Ammonia
The activation strain model and energy decomposition analysis diagrams of the boraneÀ ammonia adduct are shown in Figure 2. From Figure 2a, it can be easily seen that the energy profile in ΔE curve along the newly forming BÀ N bond is determined by the interaction energy ΔE int , which becomes destabilizing only at very short BÀ N bond distance (smaller than r BÀ N < 1.230 Å). The strain energy ΔE strain , on the other hand, becomes increasingly destabilizing as the internuclear distance decreases. The destabilizing ΔE strain stems mostly from the deformation of the Lewis acid, BH 3 , from its planar equilibrium geometry to the pyramidal geometry it adopts in the complex. Note that the BH 3 strain energy curve DE strain;BH 3 coincides with the total strain energy curve ΔE strain , whereas the NH 3 strain energy curve DE strain;NH 3 is flat all along the reaction coordinate.
Since the interaction energy plays a critical role on the formation of the H 3 BÀ NH 3 Lewis pair, we further decomposed ΔE int into four physically meaningful terms according to Eq. (3). The results of this energy decomposition analysis (EDA) are shown in Figure 2b. This graph shows us a quite straightforward picture. The ΔE int is equally stabilized by orbital and electrostatic interactions, the ΔE oi and ΔV elstat curves nearly coincide at all BÀ N bond distances shown. Both terms become more stabilizing as the fragment separation decreases and the bond begins to form, because of the increase in both HOMOÀ LUMO orbital overlap and charge penetration of nuclei with electron clouds. The stabilizing effect of ΔE oi and ΔV elstat is, however, opposed by the Pauli repulsion ΔE Pauli term. Note that at a BÀ N separation shorter than the equilibrium bond length, the upward slope of the ΔE Pauli curve is larger than the downward slope of the ΔE oi and ΔV elstat curves, which is the reason behind the destabilization of ΔE int at short internuclear distance. [31] The dispersion energy ΔE disp , on the other hand, remains nearly constant at any point along r BÀ N .
Thus, electrostatic and orbital interactions are the main contributors to the formation of the H 3 BÀ NH 3 Lewis pair. To understand the origin of the stabilizing ΔE oi and ΔV elstat , we have analyzed the molecular orbital (MO) diagram of the fragment molecular orbitals (FMOs) and the electrostatic potential surface of each fragment, respectively. [26a] To ensure that our results are not skewed by the fact that the Lewis  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57 adducts have different equilibrium bond lengths, analysis of all Lewis pairs will be performed at the same r BÀ N distance of 1.687 Å, near to the equilibrium bond distance of boraneÀ ammonia. Energies at consistent geometry for the H 3 BÀ NH 3 adduct are shown in Figure 2c. Figure 3a shows that ΔE oi can be rationalized in terms of the well-known [HOMO(base)À LUMO(acid)] interaction between the  filled N 2p z orbital of the Lewis base with the empty B 2p z orbital of the Lewis acid (see Figure 3b). This interaction has favorable orbital energy gap (Δɛ = 2.5 eV) and overlap (hHOMO j LUMOi = 0.36). [32] Furthermore, inspection of the electrostatic potential surfaces illustrated in Figure 3c and atomic charges in Figure 3d reveals that accumulation of positive charge around the boron atom of the electron-deficient Lewis acid and negative charge around the nitrogen atom of the electron-rich Lewis base are responsible for the stabilizing ΔV elstat .
In summary, the EDA along the forming H 3 BÀ NH 3 Lewis pair demonstrates that the attractive interaction between the BH 3 Lewis acid and the NH 3 Lewis base has a stabilizing covalent character that is the same magnitude as the electrostatic character, both can be easily understood in terms of simple chemical arguments. Our results, so far, conform to and agree with the current picture presented in the literature. [6d,e,i] In the coming next sections, we extend our analysis to study the stability of Lewis adducts of halogenated Lewis acids and Lewis bases.

Halogenated Lewis Acids
Next, we turn to the analysis of the formation of the Lewis pairs between boron trihalides and ammonia. The activation strain model and energy decomposition analysis diagrams for the X 3 BÀ NH 3 Lewis pairs (where X = F, Cl, Br, and I) are shown in Figure 4. In line with the expected Lewis acidities, [9c] BI 3 forms the strongest complex with ammonia and the energy of formation of the Lewis adduct ΔE decreases in strength, i. e., becomes less stabilizing, along the series: BI 3 , BBr 3 , BCl 3 , BF 3 . However, in contrast with the commonly accepted view of Lewis acid/base interaction, the stronger bond energy does not originate from the more stabilizing interaction energy, but from  the less destabilizing strain energy. [12a,33] In general, ΔE strain is less destabilizing for the Lewis complex involving BI 3 and becomes increasingly destabilizing along the series BI 3 < BBr 3 < BCl 3 < BF 3 . On the other hand, ΔE int is nearly the same for all Lewis adducts and does not follow a systematic trend. If covalent interactions would be the decisive factor for the observed Lewis pair stabilities, one would expect that the trend in ΔE int along the boron trihalides also hold for the trend in ΔE; but this is not the case. We discuss these findings in more details below.
The same conclusion can be drawn at consistent geometries (r BÀ N = 1.687 Å, see Table 1; the EDA data is given in Table S2). The values of ΔE int are of the same order of magnitude as in boraneÀ ammonia, ca. 41 kcal mol À 1 , while the ΔE strain is significantly larger for the boron trihalides and accounts for 12.2 and 22.3 kcal mol À 1 for H 3 BÀ NH 3 and F 3 BÀ NH 3 , respectively. The ΔE strain results predominantly from the deformation of the Lewis acid DE strain;BX 3 . Nevertheless, there is no clear correlation of DE strain;BX 3 with any geometrical change. The pyramidalization angle Dq pyr;BX 3 is very similar for all Lewis acids and the BÀ X bond stretching Δr BÀ X has a reversed trend that from DE strain;BX 3 , i. e., the Δr BÀ X increases as X goes from F to I (see Table 1).
In order to pinpoint the origin of the observed strain energy of the boron trihalides, we have carried out a subsequent analysis on the BX 3 fragment. This time we decompose the DE strain;BX 3 term into the individual strain energies associated with the bending of the θ XÀ BÀ X angle (ΔE strain,θ ) and the BÀ X bond stretch (ΔE strain,r ), as schematically illustrated in Table 2. First, the BX 3 is pyramidalized with a fixed r BÀ X , taken from the respective planar equilibrium geometry, and, next, the r BÀ X bond is allowed to relax to the one it has in the consistent geometry  Table S2 for data at consistent geometries. of the Lewis pair. The energy associated with each geometrical deformation is presented in Table 2. The majority of the strain energy originates from the bending of the θ XÀ BÀ X angle and the trends in ΔE strain,θ follow exactly the trends of the total strain of the Lewis acid DE strain;BX 3 , that is, it is larger for BF 3 and smaller for BI 3 . The same trend can be observed if we analyze the other way around, first elongation of the r BÀ X bond and then bending of the θ XÀ BÀ X angle (see Figure S2 and Table S3). Yet, the pyramidalization angle is similar for all boron trihalides. Why then does BX 3 become easier to bend to the same extent along the series X = F, Cl, Br, I? To answer to this question, we must understand exactly how the electronic structure of the Lewis acid changes upon pyramidalization (that is, bending and elongation). The rise in energy associated with the deformation of BX 3 (i. e., DE strain;BX 3 ) could stem from two distinct factors: i) the bonding between central boron and halogen ligands becomes less stabilizing in the pyramidal geometry; and ii) there is an increase in the repulsion among the halogens as BX 3 deforms. [34] Therefore, we have further decomposed the DE strain;BX 3 in terms of the interaction energy between B and X 3 (DE int;BÀ X 3 ) and among the three X (ΔE int,XÀ XÀ X ), more specifically, in terms of the change in both energy terms as BX 3 deforms from the planar to the pyramidal geometry (Table 3; see supporting methods for a complete derivation).
Put simply, the interaction energy ΔE int,XÀ XÀ X corresponds to the formation of the (X * ) 3 fragment in its quartet valence configuration and in the geometry which it acquires in the overall molecule, and the interaction energy DE int;BÀ X 3 corresponds to the actual energy change when the prepared B-sp 2 and (X * ) 3 fragments are combined to form the BX 3 (planar or pyramidal). As BX 3 goes from one geometry to the other, the change in interaction energy is written as ΔΔE int . Thus, the ΔΔE int,XÀ XÀ X and DDE int;BÀ X 3 are, respectively, the change in both interaction energy terms when BX 3 goes from the planar to the pyramidal geometry and sum to DE strain;BX 3 (see Table 3). Here, positive values of ΔΔE int indicate that the interaction energy opposes pyramidalization, while negative values indicate that it favors pyramidalization of the Lewis acid.
The most striking result in Table 3 is that the interaction energy between the halogens, which is predominantly repulsive (see Table S4), becomes less destabilizing in the pyramidal geometry (i. e., ΔΔE int,XÀ XÀ X is negative) and, thus, favors the pyramidalization of the Lewis acid. This is because when the r BÀ X bond elongates, the halogens are actually farther removed from each other in the pyramidal than in the planar geometry (see Δr XÀ X in Table S3). This means that DDE int;BÀ X 3 determines the trends in DE strain;BX 3 , as clearly observed from Table 3. Along X = F to I, DE strain;BX 3 varies from 22.2 to 15.3 kcal mol À 1 and DDE int;BÀ X 3 varies from 27.0 to 20.4 kcal mol À 1 . The interaction Table 2. The strain energy terms (in kcal mol À 1 ) associated with the stepby-step deformation of the Lewis acid from the planar to the pyramidal geometry. [ [a] Geometry adopted in the complex with a BÀ N distance of 1.687 Å of the X 3 BÀ NH 3 Lewis pairs (where X = F, Cl, Br, and I), computed at ZORA-BLYP-D3(BJ)/TZ2P. Table 3. Change in the energy decomposition analysis terms (in kcal mol À 1 ) associated with the deformation of the BX 3 Lewis acids from the planar to the pyramidal geometry [a] (where X = F, Cl, Br, and I). [

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energy between the boron and the halogens is less stabilizing in the pyramidal than in the planar geometry (i. e., DDE int;BÀ X 3 is positive) and, thus, opposes the pyramidalization of the Lewis acid. This loss in stabilization correlates to the difficulty to pyramidalize the Lewis acid, that is, a larger DDE int;BÀ X 3 translates in to a larger DE strain;BX 3 . To obtain insight into the different contributors to the interaction energy we have again employed the EDA scheme [27] (see Table 3, full data is provided in Table S4). It can be seen that the trend in DDE int;BÀ X 3 is dictated by the orbital interactions DDE oi;BÀ X 3 . Both DDE int;BÀ X 3 and DDE oi;BÀ X 3 oppose pyramidalization of the Lewis acid (i. e., they are positive) and decrease in magnitude from BF 3 to BI 3 . Along X = F to I, DDE oi;BÀ X 3 varies from a value of 94.7 to 33.2 kcal mol À 1 . Note that the electrostatic interaction also opposes pyramidalization (i. e., positive values of DDE elstat;BÀ X 3 ) but it increases from BF 3 to BI 3 , therefore, not following the trend in DDE int;BÀ X 3 . Interestingly, the Pauli repulsion term favors pyramidalization (i. e., negative values of DDE Pauli;BÀ X 3 ) because it goes with an elongation of the r BÀ X bond in the pyramidal geometry, which becomes longer from BF 3 to BI 3 . Therefore, in a sense, DDE elstat;BÀ X 3 and DDE Pauli;BÀ X 3 work together against the observed trend in DDE int;BÀ X 3 . Finally, the dispersion term, ΔΔE disp,BÀ X3 , is the same at both geometries (i. e., DDE disp;BÀ X 3 = 0.0 and is not provided in Table 3). Figure 5 shows the MO diagram of the main orbital interactions between (X * ) 3 and B-sp 2 in the e 1 and a 1 representations (the complete MO diagram with all valence orbitals is provided in Figure S3). We now address why the covalent component of the interaction between B and X 3 is less stabilizing in the pyramidal geometry, that is, DDE oi;BÀ X 3 is positive, and how it determines the trend in DDE strain;BX 3 . Most of this effect originates from the orbital interactions in the e 1 irreducible representation (see Table 3), which corresponds to the bonding in the plane of the molecule (σ-like bonding). Interestingly, the total stabilizing orbital interactions DDE oi;BÀ X 3 is provided by nearly 70% DDE oi;e 1 and 30% DDE oi;a 1 (the contribution from DE oi;a 2 is very small, see Table S4). This is in contrast to the common belief that the strength of the BÀ X bond arises from the overlap in the π system (i. e., in the a 1 representation), between the filled np z orbitals of the halogens and the empty p z orbital of boron. [9c] In planar BX 3 (Figure 5a left), two electron pair bonds are formed in the e 1 irreducible representation (ne 1 � 2p x and ne 1 � 2p y ), where ne 1 is a combination of the np orbitals in the xy plane of the halogen atoms. The degenerated singly occupied ne 1 orbitals show the well-known increase in energy on descending group 17 in the periodic table, [35] from À 12.7 to À 8.0 eV as X goes from F to I, associated with the decreasing electronegativity of X. [36] As the fragments combine to form BX 3 , the electrons are stabilized in the bonding molecular orbitals and this stabilization correlates well with the energy of the (X * ) 3 fragment orbitals, in line with the order of strength of the BÀ X bond. [37] Upon pyramidalization, there is a decrease in the orbital overlap between ne 1 and 2p x,y for all BX 3 (see Table S4), resulting in the less stabilizing DDE oi;BÀ X 3 . Furthermore, pyramidalization also results in destabilization of the ne 1 orbitals of the (X * ) 3 fragment and, most important, in the bonding molecular orbitals of BX 3 (Figure 5a right). Interestingly, the destabilization of the bonding molecular orbitals shows the same trend as the DE strain;BX 3 , that is, it decreases along the series X = F, Cl, Br, I (Δɛ = 0.4, 0.3, 0.3, and 0.2 eV for BF 3 , BCl 3 , BBr 3 , and BI 3 , respectively). Similar effect occurs for orbital interactions in the a 1 representation (see Figure 5b). Thus, as the boron trihalides deform to the same extent, the destabilization in the molecular orbitals of BF 3 is larger. The (F * ) 3 is more strongly bound to the central boron atom, therefore, the decrease in the strain energy from BF 3 to BI 3 can be ascribed to the amount of energy required to distort a weaker bond. In other words, it requires less energy to deform BI 3 than BF 3 because the BÀ I bond is weaker than the BÀ F bond.
At last, we comment on the role of the orbital interactions between the Lewis acid and the Lewis base to the stability of the X 3 BÀ NH 3 Lewis pairs (where X = F, Cl, Br, and I), which is the widely accepted rationale to explain the Lewis acidity of boron trihalides. [11] Our EDA results ( Figure 4b and Table S2), indeed, demonstrate that ΔE oi follows the trend in ΔE, that is, it becomes more stabilizing from F 3 BÀ H 3 to I 3 BÀ NH 3 . The trends in ΔE oi can be ascribed to the energy of the LUMO of BX 3 that decreases in energy from BF 3 to BI 3 (see Figure 5b), resulting in more stabilizing orbital interactions, in line with the results by Bessac and Frenking. [11b] However, the stabilizing effect of ΔE oi (and also ΔV elstat ) is counteracted by a strong Pauli repulsion ΔE Pauli that leads to a similar ΔE int for all Lewis adducts (see Figure 4b). We again emphasize that it is crucial to compare the Lewis adducts at a consistent geometry, that is, the same r BÀ N bond length, because the energy components are highly dependent on the bond distance. [26a] Data at the equilibrium geometries (Table S1) shows that the strain energy of BF 3 is smaller than BCl 3 , but this is just because of the longer r BÀ N bond distance in the Lewis pair with the former. Analysis at the consistent geometries (Table 1) shows that the trends in bond energy ΔE can solely be assigned to the strain energy of the Lewis acid DE strain;BX 3 ; not to the interaction energy ΔE int .
We conclude that the more destabilizing strain energy along the series BI 3 < BBr 3 < BCl 3 < BF 3 , leads to less stable X 3 BÀ NH 3 Lewis pairs (where X = F, Cl, Br, and I), due to a loss in stabilization of the bonding interactions between the central boron and the halogen ligands as the BX 3 goes from the planar to the pyramidal geometry. This effect is most pronounced for BF 3 because the BÀ F bond is the strongest in our series of boron trihalides. These general observations also explain why a reversed trend is observed for the interaction of boron trihalides with weak bases: [9] weak bases induce small distortion of BX 3 from its planar equilibrium geometry that allows the interaction energy to dominate and govern the bonding of these Lewis pairs.

Halogenated Lewis Bases
Finally, we turn our attention to the formation of Lewis adducts between borane and nitrogen trihalides. The activation strain model and energy decomposition analysis diagrams for the H 3 BÀ NY 3 Lewis pairs (where Y = F, Cl, Br, and I) are shown in Figure 6, whereas data at consistent geometries is summarized in Table 4. The NI 3 forms the strongest complex with borane and the energy of formation of the Lewis adduct ΔE decreases in strength, i. e., becomes less stabilizing, along the series: NI 3 , NBr 3 , NCl 3 , NF 3 . Trends in ΔE curves originate solely from a more stabilizing interaction energy ΔE int . Note that the strain energy ΔE strain curves show a reversed trend, overruled by the trend in ΔE int , namely, NF 3 has a less destabilizing ΔE strain than NI 3 . Therefore, similar to boraneÀ ammonia, the relative stability of the H 3 BÀ NY 3 Lewis pairs is determined by ΔE int .  The observed trend in ΔE int curves is given by the orbital interaction ΔE oi curves, that is most stabilizing for the Lewis complex with NI 3 and decreases in strength along the series NI 3 , NBr 3 , NCl 3 , NF 3 . From NI 3 to NF 3 , at the consistent geometry (see Table 4), ΔE int varies from a value of À 28.3 to À 16.9 kcal mol À 1 and ΔE oi varies from a value of À 71.5 to À 63.0 kcal mol À 1 . This is paralleled by a decrease of Pauli repulsion, that varies from a value of 99.4 to 88.2 kcal mol À 1 from NI 3 to NF 3 , as reflected by the decreasing number of core electrons and diffuse orbitals as the halogen decreases in size. Trends in ΔV elstat , on the other hand, are not exactly systematic along the Lewis bases. They are partially inverted and decreases in strength along the series NI 3 , NCl 3 , NBr 3 , NF 3 . Finally, the dispersion energy ΔE disp has the smallest contribution to ΔE int (not shown in Figure 6, see Table 4 for data at consistent geometries).
Formation of the H 3 BÀ NY 3 Lewis pairs involves a key orbital interaction between the filled out-of-phase mixing of N 2p z and halogens np z orbitals of NY 3 with the empty B 2p z orbital of BH 3 , the HOMO(base)À LUMO(acid) interaction (see Figure 7; additional stabilizing contribution from the HOMO-2(base)À LUMO (acid) interaction is given in Figure S4). However, this interaction is relatively less stabilizing compared to boraneÀ ammonia. As the Y ligands vary from H to the increasingly more electronegative atoms I, Br, Cl and F, the HOMO drops in energy, which leads to a larger HOMOÀ LUMO energy gap (Δɛ HOMOÀ LUMO = 2.5, 2.8, 3.4, 3.8, and 5.5 eV along NY 3 = NH 3 , NI 3 , NBr 3 , NCl 3 , and NF 3 , respectively). The corresponding orbital overlap hHOMO j LUMOi, on the other hand, decreases on descending group 17, i. e., it becomes less favorable. Because of the out-of-phase mixing of the np z orbitals, the amplitude of HOMO is larger on the less electronegative atom (either N or Y). Thus, the amplitude on the nitrogen atom decreases along the series NF 3 , NCl 3 , NBr 3 , and NI 3 , which decreases the spatial overlap with the empty 2p z orbital of BH 3 . Therefore, the trend in Δɛ HOMOÀ LUMO overrules the trend in hHOMO j LUMOi, determining the trend in orbital interaction energies and, eventually, in the stability of the H 3 BÀ NY 3 Lewis pairs.

Conclusions
At variance with the current view, the strength of archetypical X 3 BÀ NY 3 Lewis pair (where X,Y = H, F, Cl, Br, and I) bonds is not solely attributed to the strength of the stabilizing frontier molecular orbital interactions. The bonding mechanism involving boron trihalides, for example, is determined by the amount of energy required to deform the fragments, especially the Lewis acid, upon complexation. This follows from our detailed bonding analyses based on relativistic dispersion-corrected density functional theory at ZORA-BLYP-D3(BJ)/TZ2P.  Table 4 for data at consistent geometries.
Our activation strain and quantitative KohnÀ Sham MO analyses reveal that the bonding energy of the series X 3 BÀ NH 3 is determined by the strain energy associated with the geometrical distortion of the Lewis acid on going from the planar to the pyramidal geometry acquired in the Lewis complex. We have, for the first time, quantitatively decomposed the strain energy of the Lewis acid in terms of the change in the interaction energy within one fragment upon its deformation. The decrease in the strain energy directly correlates with the weakening of the BÀ X bond as the electronegativity of X decreases along the series: F, Cl, Br, and I. Most of this effect arises from the bonding in the plane of the molecule, not in the π system as is widely believed. In other words, the less destabilizing energy required to deform a weak BÀ X bond results in a smaller strain energy, which manifests in a more stable Lewis pair. This is the actual reason why the Lewis pairs becomes systematically stronger as BX 3 goes from BF 3 to BI 3 , and not because of a more stabilizing interaction energy as is the currently accepted rationale. For the H 3 BÀ NY 3 series, the bonding is driven by the charge-transfer stemming from the commonly accepted HOMOÀ LUMO interaction between the lone pair on the nitrogen of the Lewis base and the empty p orbital at the boron of the Lewis acid.
This work clearly demonstrates the role of the strain energy, besides the well-known donorÀ acceptor orbital and also electrostatic interactions, in playing a leading role in determining the strength of Lewis acid/base interactions. Our findings are both chemically intuitive and grounded in quantum chemical findings based on state-of-the-art computations. Importantly, we have brought our understanding of these fundamental interactions into the 21 st century and hope that this work will be useful for the development of novel Lewis pair chemistries .  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57