The nature of multiple boron-nitrogen bonds studied using electron localization function (ELF), electron density (AIM), and natural bond orbital (NBO) methods

Local nature of the boron-nitrogen (BN) bonding with different formal multiplicities (B≡N, B=N, B-N) have been investigated for 25 experimentally established organoboron molecules in both real and the Hilbert space, using topological analysis of electron localization function (ELF), electron density (AIM), and natural bond orbital (NBO) method. Each BN bond has been represented (ELF) by the bonding disynaptic attractor V(B,N), with the basin electron population between 5.72e and 1.83e, confirming possible existence of all the three bond types. A covalent character of bonding can be associated with the dative mechanism due to the V(B,N) bonding basin formed mainly (91–96%) by the N electron density. Similarly, the NBO method shows 2-center natural orbitals, consisting largely of the hybrids from the N atom. The AIM analysis yields the features typical for shared (H(3,−1)(r) < 0) and closed-shell (∇2ρ(3,−1)(r) > 0) interactions. The delocalization indices, describing electron exchanges between B and N quantum atoms, are smaller than 1.5, even for formally very short triple B≡N bonds. Graphical abstract . Electronic supplementary material The online version of this article (10.1007/s00894-020-04374-9) contains supplementary material, which is available to authorized users.


Introduction
The nature of the boron-nitrogen bond (BN) partially determines the physico-chemical properties of molecules, with growing importance in chemical synthesis and applications. Compounds with BN chemical bonds are used to prepare edge-rich, B and N dual-doped graphene sheets (BNGs) to ensure more superior electro-catalytic activity for oxygen reduction [1,2]. In supramolecular chemistry, scientists can construct crystalline and soft molecular networks using dative BN bonds [3]. The molecular cages containing six dative boron-nitrogen bonds can encapsulate polyaromatic molecules such as triphenylene [4]. Recently, Légaré et al. [5] showed that catenation of two N 2 molecules (formation of nitrogen chains) under near-ambient conditions is possible using organoboron compound. In the postulated reaction between dinitrogen and a hypothetical base-stabilized borylene, the dipotassium complex is formed with the B ... N ... N ... N ... N ... B chain [5]. In an earlier study, Légaré et al. [6] reported that the reaction of N 2 molecules bound to two borylene units results in the species with the neutral (B 2 N 2 ) or dianionic [B 2 N 2 ] 2− fragments. Such reactions are important for the development of the efficient process of nitrogen fixation and reduction by boron. Zhang et al. [7] demonstrated that the BN bonds, forming the hexagonal boron nitride nanosheet, play an essential role in electrochemical catalysis of N 2 to NH 3 . It has been postulated that unsaturated boron on the edge can activate inert dinitrogen molecule. In the world of fluorescent organic compounds, incorporation of the B-N bond within the polycyclic aromatic hydrocarbons is a method of rigidifying the compound core, yielding more intense fluorescence, extraordinary thermal and photochemical stability, and high fluorescence quantum yields. For example, Saint-Louis et al. [8] when studying polycyclic azaborine chromophores, designed and synthetized the molecule, where the N-C(=O)-N unit has been replaced with the N-B(OH)-C unit. The obtained This paper belongs to the Topical Collection Zdzislaw Latajka 70th Birthday Festschrift Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00894-020-04374-9) contains supplementary material, which is available to authorized users. azaborine molecules displayed higher molar absorption coefficient than their carbonyl analogs and presented higher emission quantum yields when compared with the imide analogs. In summary, exploration of the BN bond properties is important for the development of new organic field-effect transistors, solid-state lasers, biological imaging, or organic lightemitting diodes.
Discussion of molecular properties is very often related to the assumption that atoms in a molecule are connected by chemical bonds. Thus, their length, type, strength, and applied electronic descriptors, as bond population or the degree of delocalization of electron density (e.g., aromaticity), are crucial for finding correlations between the electronic structure and particular properties. From this point of view, knowledge of the BN bond local electronic structure is essential for predicting its physical properties.
Covalent bonds are usually characterized as single, double, triple (i.e., A-A, A=A, A≡A), or higher order according to the expected electron pairs of polarized electrons, which should be spatially localized between the atomic cores. The Lewis formula [9], introduced before quantum chemistry era, where covalent bonds with shared electron pairs are represented by lines are well accepted multiple bond representation in chemistry. Nowadays, the number of electrons in the chemical bond, localized in space between core regions, can easily be calculated using topological concepts such as attractor, attractor's basin, and separatrix with molecular distributions of electron localization function (ELF) η(r) [10][11][12][13][14][15]. Thus, the number of bonding electrons, predicted using the Lewis formula or molecular orbital theory, can be verified using modern concepts of quantum chemical topology (QCT) [16].
The Cambridge Structural Database (CSD) yielded 48,052 crystal structures with boron-nitrogen contacts between 1.230 and 2.319 Å [17]. Percentage distribution of those contacts is presented in Fig. 1. Two maxima have been observed at approximately 1.43 and 1.54 Å bond lengths, associated with molecules, where the B atom is bonded (mainly) to three and four atoms, respectively.
In the present study, we concentrated on 25 molecules, containing 37 boron-nitrogen bonds selected from the CSD, the prototypical H 3 N-BH 3 molecule with a single dative bond (N→B), and borazine molecule (B 3 H 6 N 3 ) with the delocalized B ... N bond. Such a variety ensures existence of all the three types of the BN bond, guaranteeing exploration of the nature of the B-N, B=N, and B≡N bonds. The local nature of the boron-nitrogen bonds has been investigated using two methods in the real space, i.e., topological analysis of the electron localization function, η(r) (ELF) and topological analysis of electron density, ρ(r) (AIM) [18]. Obtained results are independent of a chosen set of molecular orbitals. The third method applied, the natural bond orbital (NBO) method [19], chosen for comparison, is the analysis performed in the Hilbert space.
The main purpose of this study is to describe local electronic structure of the boron-nitrogen bond from the perspective of the topological analysis of η(r) and ρ(r) fields and to compare the results with those obtained with two-center NBOs. Furthermore, we are interested in verifying to which extent a formal representation of the bond multiplicity (B-N, B=N, B≡N), used in the formal Lewis dot formula [20], stays in agreement with the modern topological analysis of the nature of the BN chemical bond. We also looked for correlations between values characterizing BN bonds, obtained by different approaches and for various bond lengths.
The present study is a part of a larger project, comparing properties of the BB, BC [21], BO [22], BN [23] BF, BCl [24], and BCu [25] bonds, investigated by topological analysis of ELF.
The def2-TZVP and TZP basis sets have been obtained using the Basis Set Exchange software [44,45].
The minima on potential energy surface (PES) have been verified through non-imaginary harmonic vibrational frequencies. All the molecules have been studied at 0 K and at singlet electronic states.
The hydrogen atoms have been added to the yecvor molecule, obtained from the CSD before performing the geometry optimization, using the Lewis formula presented in Ref. [40]. Similarly, hydrogen atoms have been added to the cetsup [46], cofvuo [47], bpampb [48] structures, obtained from CSD.
Topological analysis of the ELF has been carried out using the TopMod program with a cubical grid of step size 0.05 bohr [49,50]. In the case of two or three V i (B,N) basins, the simple sum of their basin populations has been considered. In the case of three molecules with the single B-N bond, the monosynaptic non-bonding basin V(N) has been found in the region of the bond instead of the disynaptic basin V(B,N). Such result should be treated with caution because the classification to the synaptic type may depend strongly on numerical details and procedures applied in a program used for topological analysis.
The values of electron delocalization index (DI) in the framework of topological analysis of ρ(r) field, have been calculated using the AIMall program [51].
Graphical representations of molecules have been constructed using the JMol program. The ELF domains have been visualized using the VMD [52] and with UCSF Chimera [53], developed by the Resource for Biocomputing, Visualization, and Informatics at the University of California, San Francisco, with support from NIH P41-GM103311.
The NBO analysis has been performed using the version 3.1 of the program, incorporated in the Gaussian 16, Revision E.01 [54]. No "Resonance" and "3CBOND" keywords have been used since only two-center molecular orbitals have been searched for. Analysis has been performed for all optimized molecules at the DFT(M062x)/6-311+G(d,p) computational level.

Results and discussion
Chemical names of the studied molecules, downloaded from the CSD [17] and the experimental boron-nitrogen bond lengths, r exp (BN), are presented in Table 1. The molecules contain formal single (B-N), double (B=N), and triple (B ≡ N) bonds and cover the range of r exp (B,N) values between 1.220 (dogpiy [55]) and 1.717 Å (ajepah [56]).
We have divided selected compounds into three groups, according to the formal BN bonds, published elsewhere. The first group of compounds contains the triple boron-nitrogen bond, B≡N, the second group consists of molecules with formal double bonds, B=N, and the third group gathers compounds with formally single B-N bonds.
All the properties discussed in the text have been obtained for molecules optimized using the DFT(M062X) method and 6-311+G(d,p) basis set, unless stated otherwise.
Topological analysis of η(r) function shows three (cetsup, dogpiy) or two (sictii, vejhib) bonding disynaptic attractors V i (B,N) for the boron-nitrogen bond. According to the interpretation proposed by Silvi and Savin [11], the BN bond has a covalent character with electron density shared by both atoms. The exemplary V i = 1,2,3 (B,N) attractors in the cetsup molecule are shown in Fig. 3. The V i = 1,2,3 (B,N) attractors are found outside the imaginary line joining the B and N nuclei. This can be explained by the increased electron localization in the molecular regions with more effective overlapping of the porbitals in π-fashion. Similar spatial localization of bonding attractors have been found for the C=C bond [59]. The existence of the three valence V i = 1,2,3 (B,N) bonding attractors is, however, not an evidence that a single V(B,N) attractor corresponds to any particular (σ or π) bond. The presence of all the three attractors is associated with local symmetry, governed by threefold symmetry axis.
An additional evidence for a triple B≡N bond formation is the absence of the monosynaptic non-bonding attractors V(N), characterizing the non-bonding electron density of the N atom. Since such attractor is not observed, all valence electrons of the nitrogen atom are engaged in the formation of the B≡N and NC bonds. The conclusions above are illustrated well by cetsup. The respective localization domains, corresponding to the CC, BC, B≡N, and NC bonds are presented in Fig. 3.
In order to confirm that the studied short boron-nitrogen bonds are triple bonds, analysis of the basin populations, N , needs to be performed. Those populations are calculated for localization basins associated with the V i (B,N) attractors. The results are presented in Table 2. The total value of N ranges between 5.72 (cetsup) and 5.57e (dogpiy). Such large populations undoubtedly indicate the triple B≡N bond. It is worth noting that topological analysis of ELF for the prototypical molecule with the triple C≡C bond, acetylene, HC≣CH studied by Silvi et al. [59] yielded the total population of 5.14e for the V(C,C) basin. Thus, the value of 6e, predicted with the MO or Lewis formula is only a guidance. All bonds with the basin population of V(B,N) larger than 5e in this study are considered to possibly be a triple bond. Since the N values are slightly smaller than the formal value of 6e, the resonance equilibrium for structures with the B=N and B≡N bonds should be considered with dominant contribution of the latter.
All the studied B≡N bonds are polar, and the polarization of its electron density is associated with inequivalent character (electronegativity) of the B and N atoms. Polarity can be measured by the polarity index, p NB . The index has been proposed by Raub and Jansen [60] based on combined topological analysis of ρ(r) and η(r) fields. It assumes values between 0 for homopolar and 1 for idealized ionic bonds, respectively.
Complementary insight into the bonding nature of the B≡N bonds can be obtained from the topological analysis of electron density, ρ(r) [18]. This method is much widely known than the topological analysis of ELF. At this point, it is worth recollecting the two papers by Bader et al. [61,62] who observed that the fields of the negative Laplacian of ρ(r), −∇ 2 ρ(r), and ELF are generally homeomorphic [61] and showed that ELF has no direct relationship with the Laplacian of the conditional pair density [62].
The values of electron density for the B≡N bond critical point (BCP), ρ (3,−1) (r), and other topological parameters are shown in Table 3. As expected, the results are different than those obtained from the ELF representation. The value of the ρ (3,−1) (r) for four molecules is large (≈ 0.27 e/bohr 3 ) and typical for covalent bonds with shared electron density. However, the value of the Laplacian of ρ (3,−1) (r) for the BCP, ∇ 2 ρ (3,−1) (r), is large and positive (1.19-1.35 e/bohr 5 ) indicating a decrease of the electron density around the BCP, typical for closed-shell interactions. It shows that the B≡N bonds cannot be classed as typical covalent-polarized bonds. However, the values of the Cremer and Kraka [63,64] total energy density, H (3,−1) (r), are negative (between − 0.27 and − 0.26 au/bohr 3 ) and show prevalence of the negative potential energy, typical for the covalent bonds.
Another interesting measure of a bond character is the electron DI containing information about the average number of electrons shared between quantum atoms. Practical application of the DI in explaining the nature of atomic interactions has been described by Bader et al. [65] in the article entitled "Where To Draw the Line in Defining a Molecular Structure." In that article, the representation of the bonding (not bonds [66]) in the adduct between HSiCl 3 and Cp(CO) 2 Mn molecules have been rationalized on the basis of topological analysis of electron density. The values of DI were 0.19 for LiF (the ionic limit), 2.7 for C 2 , and 1.6 for CO molecules. For the B≡N bonds in the sictii, dogpiy, vejhib, and cetsup molecules, the values of DI are between 0.85 and 1.28. Those values are  Table 3 Data for the B≡N bond, obtained from topological analysis of electron density field (AIM) for four molecules with formal triple boronnitrogen bond. Calculations performed at the DFT(M062x)/6-311+ G(d,p) computational level   much smaller than 3, expected for an ideal triple bond, and do not confirm the triple B≡N bond existence. Calculations performed at the same computational level for the HC≡CH and N≡N molecules with triple bond yielded the DI of 2.851 and 3.046, respectively. Such results are in agreement with the formal concept of the triple bond in those molecules. In summary, the results obtained for the B≡N bonds do not confirm a triple covalent bond with 6e shared between the B and N atoms. The boron-nitrogen interaction in the studied molecules shows some similarities with the N→B bond in the H 3 N-BH 3 molecule, formally understood as covalent-dative. The signs of the Laplacian of ρ (3,−1) (r) are positive, however the value of ρ (3,−1) (r) is about two times smaller than those calculated for the B≡N bond. Such small value can indicate a bond with only 2 electrons in the bonding region, much smaller than six electrons found for the B≡N bond. Recently, a dative bond in a series of small molecules has been investigated by Mebs and Beckmann [67], presenting similar topological characteristic of the BN bond. In summary, within the topological analysis of ρ(r) field, a formally triple B≡N bond in cetsup, dogpiy, vejhib, and sictii has the topological bond order (AIM) close to 1.
Finally, B≡N bonds have been analyzed in the Hilbert space, via NBOs [68,69]. This method has been applied to search for 2-center valence natural orbitals, formed by natural atomic hybrids (h A ) at the B and N atoms. The occupancies of the NBOs, the percentage contribution of the B and N atoms to the NBOs, and sp λ composition of each natural h A are shown in Table 4. The results show that all the studied BN bonds are triple, since three twocenter orbitals have been obtained. Their occupations are close to 2e (> 1.91e) and the NBOs are one σ-bond and two π-bonds. The σ-bond is formed by two sp. hybrids overlap, but two π-bonds are formed in a pure fashion by overlapping of the p-orbitals.
Each of the studied BN bonds has been characterized by the bonding disynaptic attractor V(B,N), thus the bonds have covalent-polarized character. In the cojwaa, cofvuo, sictii, zeypuo, bpampb, iditas, and notlud molecules, single V(B,N) attractors are observed, while in the yecvor and axuviy molecules, pairs of V i = 1,2 (B,N) attractors are localized for two BN bonds. Such local topology of ELF is undoubtfully associated with planar geometrical structures of both molecules.
Values of the BN basin populations and other topological parameters obtained from topological analysis of ELF are presented in Table 5. The largest population of 4.54e have been obtained for the shortest (1.255 Å) BN bond investigated in this series of the molecules, i.e., the cojwaa molecule cis,mer-[(Br) 2 (Me 3 P) 3 Rh(B≡NSiMe 3 )]. The BN bond has an interim character between a double (dominant) and a triple bond, since the value of N for V(B,N) is larger than a formal value of 4e. Topological analysis of ELF done through localization of the non-bonding attractor and basin V(N) does not show any lone pair on the nitrogen atom. Thus, all valence electron density of the N atom is engaged in formation of bonds with the neighboring atoms, including the NSi bond. This bond, formally single, with the basin population of 3.17e have an interim character between N-Si and N=Si. On the  Similar to other studied compounds with triple B≡N bonds, the bond is heavily polarized towards the N atom.
In the cofvuo molecule, (iPrBNtBu) 3 , two B=N bonds are the shortest of seven B-N bonds and are expected to be of double character (see Scheme 1). The molecule has been synthetized by Paetzold et al. [47] and constitutes an interesting case of six-atom cluster, consisting of three boron and three nitrogen atoms. Its optimized structure is shown in Fig. 5 a. The trimer (iPrBNtBu) 3 is formed in a process of trimerization of the iPrBNtBu iminoborane [47]. According to Paetzold et al. [47], the molecule has electronic structure resembling borazine with Dewar-type 2-center 2-electron (2c-2e) localized bonds inside the cage. The experimental study [47] showed that peripheral BN bonds are short (1.364 Å, 1.384 Å) and can be characterized as double B=N bonds.   The sictii molecule [58] has two boron-nitrogen bonds (see Fig. 2) and its second longer bond (Δr opt (B,N) = 0.134 Å) has been considered to have a potentially double character. The shorter BN bond described above has been identified as a triple B≡N bond-this result is in agreement with the interpretation proposed by Rivard et al. [58]. The core and valence attractors in sictii are shown in Fig. 6 a. The optimized B=N bond length is 1.389 Å, a similar value to that obtained for double bonds in the cofvuo molecule (1.384, 1.390 Å). Therefore, a similar value of the N for the V(B,N) basin, close to 4e, could be expected. However, the calculated value is 2.88e, much smaller than 3.63 and 3.50e obtained for the BN bonds in the cofvuo molecule. Such result is unexpected, but can easily be explained using the ELF topology. The Lewis formula implies a trivalent N atom, thus two electrons should form a lone pair in its vicinity. Indeed, the monosynaptic non-bonding attractor V(N) is localized (see Fig. 6) near the C(N) core basin. The basin population of V(N) is 1.57e, thus not all valence electrons of the N atom are engaged in the formation of the BN and NC bonds. A relatively small (with respect to 4e) population of the BN bond (2.88e) is caused by separation of the lone pair, therefore not participating in the formation of the dative bond to the B atom. To sum up, topological analysis of ELF partially supports the -N≡B-N(C 2 ) representation of the chemical bonds in the sictii molecule with the lone pair at the N atom.
Polarity of the BN bond, measured by the p NB index value, is relatively small, 0.77. This is the smallest value calculated for all the BN bonds investigated in this series of the molecules. A total of 89% of the electron density in the V(B,N) basin comes from the N atom.
The yecvor molecule is an interesting case of the organoboron molecule with two boron-tin (BSn) chemical bonds [40]. Synthesis of yecvor has been described by Frankhauser et al. [40] who characterized the compound using the Lewis formula with two B=N bonds (see Scheme 1). The optimized bond length of both BN bonds (1.393 Å) is only slightly (Δr = 0.003 Å) longer than one of the B=N bonds in the cofvuo molecule, therefore a similar local nature of the bonding is expected. The core and valence attractors of the ELF field are shown in Fig. 7. Two B=N bonds are characterized by the pairs of bonding disynaptic attractors, V i = 1,2 (B,N). Total population of two V i = 1,2 (B,N) basins is 4.01e therefore indicating the double B=N bond. Thus, the topological analysis of ELF results confirms the classical interpretation displayed by the Lewis formula. The polarity index, p NB , of 0.88 is similar to the polarity indices obtained for previously investigated molecules (see Table 5). The results show that the bond is dominated by electron density from the N atom, donating (in total) 3.77e (94%) of the bond's electron density (basin population).
Particular attention should be paid to the three bonds in yecvor, i.e., the double C=C bond and two, formally single B-Sn bonds. The C=C bond is characterized by two bonding disynaptic attractors V i = 1,2 (C,C) localized below and above the plane. Such local topology of ELF has been observed for the C 2 H 4 molecule [12,59] and is associated with locally planar (approximately) arrangement of two C and two B atoms. However, the total basin population is 3.12e, which is essentially less than 4e expected for the double bond. This can be explained by the resonance forms with the C-C and C=C bonds. Basin populations of two V(B,C), corresponding to the two B-C bonds, formally single, are close to 2e (2.34e), thus the topological analysis of ELF supports the classical picture of the single bond. The BSn bonds are described by single bonding disynaptic attractors, V(B,Sn), with basin populations of 2.45e. The bond is of covalent-polarized type and is formed by 1.34e (55%) from the Sn atom and 1.09e from the B atom. There is a large polarity difference between the B=N bonds with p NB of 0.8 and the B-Sn bond with p SnB of 0.1.
Each N atom in axuviy forms four formally covalent bonds (2xC-N, B=N), therefore the non-bonding electron density (lone pairs) should not be observed in their vicinity. This is  confirmed by the topological analysis of η(r) function, showing lack of non-bonding V(N) attractors. Thus, the whole valence electron density of the N atom is engaged in formation of chemical bonds. The B=N bonds are described by two pairs of the bonding attractors V i = 1,2 (B,N). The total population of two V i = 1,2 (B,N) basins is 4.00e confirming existence of a double B=N bond. The polarity index, p NB , is 0.89, similar to that of the B=N bonds in the yecvor molecule.
Another type of the boron-nitrogen bond has a nature associated with electron delocalization in the atomic ring. Such bonds can be found in the iditas molecule [72] (see Scheme 1). The molecule consists of the borazine ring, B 3 N 3 with three trimethyl (Me), and two dimethyloamine groups bound to the N and B atoms, respectively. Optimized geometrical structure is shown in Fig. 4 and Fig. S1. A single nitroxy group, -ONO 2 , is bound to the boron atom by the BO bond. The molecule is a borazine derivative, and a set of analogous mesomeric structures can be proposed (see Scheme 2). Two Lewis structures, II and III, suggest the presence of three double B=N bonds and three single B-N bonds in the B 3 N 3 ring, yielding a formal BN bond order of 1.5. On the other hand, the structure I suggests a single type of the B-N bonding and a lone pair on each the N atom, which should be associated with the V(N) attractor. Optimized bond lengths in the ring (see Fig. S1) are in the range 1.415-1.455 Å. The shortest BN bonds (1.415, 1.416 Å) are localized close to the BONO 2 molecular fragment. Two r opt (B,N) bond lengths in the (CH 3 ) 2 NB groups are 1.436 and 1.435 Å, those values lie between the shortest and longest bond of the borazine ring.
The core and valence attractors for the iditas molecule are shown in Fig. 8 a. Topological analysis of ELF shows that six B-N bonds in the borazine ring and two BN bonds of the (CH 3 ) 2 NB fragments are described by single attractors and basins V(B,N). No pairs of the disynaptic V i = 1,2 (B,N) attractors have been found. Such results support delocalized character of the bonding in the ring. For benzene, the prototypical delocalized system, similar single attractors, V(C,C), are observed [11]. In order to find a difference between the iditas and the unsubstituted borazine molecule, topological analysis of η(r) function has been performed for borazine. The core and valence attractors for both molecules are compared in Fig. 8 a and  Population values for the V(B,N) basins in iditas vary from 2.90 to 4.30e. It is worth emphasizing that the bonds in the B 3 N 3 ring have smaller population (2.90-3.15e) than single (formal) B-N bonds in the (CH 3 ) 2 NB fragments (4.29, 4.30e). Thus, the topological bond order for the bonds in the B 3 N 3 ring is between 1.45 and 1.56. This result indicates delocalized B ... N bonds with a formal bond order of 1.5. Each BN bond in borazine has the basin population of 2.88e with the topological bond order of 1.44e-the values are smaller than those obtained for the iditas. The presence of the -NH 2 , -CH 3 , and -ONO 2 groups in the iditas significantly perturbs the electronic structure of the B 3 N 3 ring, increasing the population of the BN bonds. Such increase of the basin population can be associated with electron-withdrawing inductive effect of the -ONO 2 group and electron-releasing inductive effect of the -CH 3 and -NH 2 groups [74].
Two BN bonds in the (CH 3 ) 2 NB fragments have topological bond order of 2.15 and therefore show a double character. A relatively large amount of electron density localized between the C(B) and C(N) core regions can be associated with the V(N) basin absence (lone pair) on the N atom, and its electron density localized mainly in the boron-nitrogen bond. It is an interesting finding, because the formal Lewis formula does not predict the double character of the N=B bond between the (CH 3 ) 2 N group and the B atom of the borazine ring.
For the BN bonds in the iditas borazine ring, the polarity indices are between 0.81 and 0.84. Slightly larger polarities (0.88) have been obtained for the B=N bonds in the (CH 3 ) 2 N=B fragments. All the B ... N bonds in the borazine The BN bond in the next three molecules, i.e., zeypuo [71], bpampb [48], and notlud [73] is formally represented by a double B=N bond. The value of r opt (B,N) is between 1.401 and 1.424 Å, and r exp (B,N) ranges between 1.389 and 1.427 Å. The topological analysis of η(r) function shows single disynaptic bonding attractor V(B,N) for each BN bond. Basin populations for the BN bonds are as follows: 3.84e (zeypuo), 4.07e (bpampb), and 3.68e (notlud), respectively. Thus, for the longest B=N bond in the notlud molecule, the smallest basin population has been obtained. On the other hand, the B=N bond in bpampb, is longer than its equivalent in zeypuo and exhibits a larger value of N than that calculated for the shorter bond. The V(N) non-bonding attractor has not been found in any of the three molecules confirming therefore lack of lone pairs on the respective N atoms. Thus, all the valence electron density of N atom is engaged in the formation of covalent bonds. The boron-nitrogen bonds in the studied molecules can be characterized as double B=N bonds, although a single B-N bond, contribution to the mesomeric equilibrium has to be also considered. Like for other investigated molecules, the B=N bonds are formed almost entirely by electron density from the N atom, donating 3.63e (zeypuo) (95%), 3.80e (bpampb) (94%), and 3.46e (notlud) (94%) to the V(B,N) bonding basin.
The last molecule in the series is the abitud molecule. Its structure has been described by Saint-Louis et al. [8] who reported a formal single B-N bond as a part of the sixmember ring (see Scheme 1). The molecule contains an unusual N-BOH fragment [8] and intramolecular hydrogen bond, C=O ... H-O. Furthermore, for the central ring containing the B-N bond, some electron delocalization has been suggested, bound to influence the BN bond properties [8]. Thus, some double character of the bond is likely to be present, and therefore it is worth including the abitud molecule here. The experimental BN bond length of 1.44 Å is very well reproduced by optimization (1.449 Å).
The BN bond in abitud is characterized by the single disynaptic bonding attractor V(B,N). The monosynaptic non-bonding attractor, V(N), has not been found, therefore there is no lone pair on the N atom. Thus, additional electron density-as compared with the formal value of 2e assuming trivalent N atom-should be localized either in the BN bond or in the two NC bonds. Such hypothesis can only be partially confirmed for the BN bond since the value of N of V(B,N) is really close to 4e (3.63e). The BN bond has an interim character between a double B=N and single B-N with a larger participation of a double character in the mesomeric equilibrium. The basin populations of 1.96 and 2.11e for the N-C bonds, indicate single bonds, as predicted by the Lewis formula. The polarity index, p NB , is 0.89, and the bond is formed by 0.20e from the B atom (5%) and 3.42e from the N atom (94%).
Numerical parameters, obtained from topological analysis of ρ(r) field, characterizing B=N bonds in the cojwaa, cofvuo, sictii, yecvor, axuviy, zeypuo, bpampb, iditas, abitud, and notlud molecules, are collected in Table 6. Each bond has been described by the BCP. Values of ρ (3,−1) (r) for the B=N bonds are in the range between 0.267 and 0.185 e/bohr 3 , are generally smaller than those for the triple B≡N bonds (0.271-0.268 e/bohr 3 ), and decrease with the bond elongation. Similar decrease in value has been found for the ∇ 2 ρ (3,−1) (r), H (3,−1) (r), and DI parameters. All the B=N bonds are characterized by positive values of the ∇ 2 ρ (3,−1) (r) in the range from 1.063 to Fig. 8 a) The core and valence attractors in the iditas molecule. b) The core and valence attractors in the borazine molecule. Basin populations are shown for selected bonds 0.400 e/bohr 5 , smaller than those for the triple B≡N bonds (1.316-1.193 e/bohr 5 ). No pattern has been observed for the ellipticity parameter with exceptionally large values (0.156, 0.158) for the sictii and iditas molecules. Total energy density, H (3,−1) (r), is negative for all the studied B=N bonds, thus the energetics of the bonds (BCP) is dominated by the potential energy density, similarly to the B≡N bonds. Delocalization indices are between 1.018 (cojwaa) and 0.420 (iditas) and do not conform to the concept of a double bond, but suggest bond multiplicity smaller than 1.
The local nature study of the B=N bonds, carried out with the NBO analysis yields the σ and π orbitals (the double B=N bonds) for the cofvuo, yecvor, and iditas molecules. For other molecules, single B-N bonds have been found, accompanied by a lone pair on the N atom, formed by the p-orbital. The occupancy of the NBOs, the percentage contribution of the B and N atoms to the NBOs, and sp λ composition of each natural atomic hybrids h A are shown in Table 7. Double B=N bonds consist of two NBOs formed mainly by the N atoms with atomic contributions of 76-82%. The results support the finding from the topological analysis of ELF, showing that the B=N bonds are formed mainly by the electron density from the N atom.

The single B-N bonds
Finally, the B-N bond has been investigated. Formally, such 2-center bond is described by 2e and labeled as a single bond. The H 3 N-BH 3 molecule has been chosen as a reference system. The molecule also serves as a dative bond prototype. The topological analysis of η(r) function, performed using the DFT(M062x) method for the 6-311+G(d,p) and aug-cc-pVTZ basis sets, yields very similar results. In the B-N bonding region, the monosynaptic V(N) basin is observed, supporting a dative-covalent bond, with the nitrogen lone pair donated to the boron atom. It is worth emphasizing that the topological analysis of ELF does not yield any disynaptic basin V(B,N); only monosynaptic V(N) basin is observed. The basin population of V(N) is 1.88e with the 6-311+ G(d,p) basis set and 1.91e with the aug-cc-pVTZ basis set, very close to a formal value of 2e. The V(N) basin is formed by 1.80e (1.83e) from the N atom and 0.08e from the B atom, with a very high polarity index of 0.91. Thus, 96% of electron density in the B-N bond region comes from the N atom. It is evident that the topological approach, based on the ELF, applied to the B-N chemical bond in H 3 N-BH 3 confirms the classical concept of the dative bond N→B.  The NBO analysis gives some more insights. The twocenter natural orbital B-N with the population of 1.993e is composed of 19% contribution from the B atom and 81% from the N atom. The bond is formed by overlapping of the natural hybrids, i.e., sp 5.18 at the B atom and sp 1.76 at the N atom.
It is interesting to find whether the dative N→B bonding, characterized by larger than 95% contribution of electron density from the N atom, can also be found in more complex molecular structures. Therefore, 14 formally single B-N bonds in 10 molecules, with the r opt (B,N) lengths between 1.547 and 1.785 Å have been investigated. The molecules, selected from the CSD, are as follows: akesug [75], cofvuo, afucin [76], amikem [77], abemez [78], acipeh [79], abemid [78], ajepel [56], and ajepah [56]. The Lewis structures for all the studied molecules are shown in Scheme 3.
Each studied B-N bond is represented by a single valence basin V(B,N) or V(N). Such diversity of topological types is observed only for relatively long bonds, formally considered single. It is evident that all studied B-N bonds are either dative or have some covalent character due to shared electron density. The topology of η(r) function, similar to that in the H 3 N-BH 3 molecule, has been observed in the afucin and acipeh molecules. In the B-N bond region, instead of the expected V(B,N) basin, a single V(N) basin has been localized.
The values of N for V(B,N) are collected in Table 8. They range between 2.90e for the akesug molecule with the shortest B-N bond (1.547 Å) and 1.83e for one of the B-N bonds in the cofvuo molecule with the longest B-N bond (1.752 Å). The bonds can be classed as single, although a contribution of double B=N bond should also be considered in a resonance equilibrium (akesug, amikem, acipeh).
More detailed analysis of ELF topology has not shown any lone pair on the N atom for any of the molecules except for afucin, where the monosynaptic non-bonding basic V(N) has been found. The basin population of V(N) in afucin is 1.33e (see Fig. 9). This value is clearly smaller than a formal value of 2e. The remainder of the electron density is found mainly in the BN bond (2.65e). Thus, small contribution of the double boron-nitrogen bonding should be also considered. Nevertheless, the localization of separated V(N) shows that such contribution is rather small.
Polarity indices, p NB , (see Table 8) are between 0.82 (cofvuo) and 0.93 (acipeh, H 3 N-BH 3 ). Their values confirm that for all the studied molecules, the B-N bond is mainly formed by electron density from the N atom, donating from 1.69e (cofvuo) to 2.72e (akesug) to the bond. Such atomic contribution is much larger than that obtained for the B atom in the range from 0.13 (afucin) to 0.19e (cofvuo). Electron density from the N atom contributes from 91 (cofvuo) to 96% (acipeh) to a single B-N. For the prototypical H 3 N-BH 3 molecule, the percentage contribution of electron density from the N atom is 96%.
The single B-N bond will be discussed using an exemplar molecule amikem (difluoro-(4-methylaminopent-3-en-2-  [77] as a complex between BF 2 and olefin. Its optimized structure is shown in Fig. 10 a. The X-ray structure of the molecule has been discussed in the context of a possible mixture of ketoamine and enolinine tautomers [77]. In the ketoamine form, the double C=O bond is present, while in the enolimine form (see Fig. 10b) the C=N bond is observed. Each of the forms, or their weighted average, should be confirmed by the ELF results. Topological analysis of η(r) function shows 37 attractors, including 11 core attractors and 26 valence attractors. All attractors are shown in Fig. 10  Boron-fluorine bonds have partially covalent character and were studied in detail by us before [24]. In the valence shell of the N atom, the non-bonding V(N) attractor is not observed, thus whole valence electron density is localized in the CN bond and B-N bonds. This is partially supported by basin population values: the CN bond of the ring is 3.16e, the BN bond, formally single, has 2.61e, but the C-N bond formed by the methyl group has 1.78e. The double C=C bond, expected in the enolimine resonance form, has the population of 3.36e. The values of N for all bonds are presented in Fig. 10 c.
Comparison of basin populations with formal numbers of electrons for the bonds clearly shows that more than one Lewis structure has to be used to describe chemical bonding in the amikem. Nevertheless, the enolimine form is partially confirmed due to the populations of the V(C,N) and V(C,C) basins, which are 3.16 and 3.36e (close to the formal value of 4e), representing double C=N and C=C bonds respectively.   The NBO analysis used to find 2-center natural orbitals revealed the B-N bonding only for eight investigated molecules (see Table 10). For the three molecules (akesug, ajepel,  ajepah), the standard procedure used has not shown any natural bond orbital between the B and N atoms (see the "Computational details" section). Occupations of NBOs, the percentage atomic contributions, and natural hybrids of atomic orbitals forming the NBOs are presented in Table 10. All identified NBOs have the population close to 2e (1.910e-1.993e). Contribution of the B atom ranges between 15 (cofvuo) and 21% (abemez), and is 5.6 and 3.8 times smaller than the contribution of the N atom. Similar to the B≡N and B=N bonds, single B-N bonds are polarized towards the more electronegative N atom.
In the final part of our study, relationships for the BN local nature parameters, different bond lengths, and formal multiplicity have been analyzed for all the investigated molecules. As has been shown above, the basin populations of the BN bonds support the formal concept of the B-N, B=N, and B≡N bonds. Furthermore, the values are decreasing from 5.72e for the cetsup molecule to 1.83e for the cofvuo molecule along with elongation of the r opt (B,N). The dependency between the N values for V(B,N) and r opt (B,N), investigated using the regression analysis with the power model, is shown in Fig. 11.
All the studied BN bonds are highly polarized with the polarity indices, p NB , between 0.77 (sictii) and 0.93 (acipeh, NH 3 -BH 3 ). Unfortunately, we have been unable to find any essential correlation between the p NB values and the r opt (B,N) bond lengths (see Fig. S2). The bonds are polarized towards the N atom, which delivers from 5.19e (cetsup, vejhib) to 1.69e (cofvuo) to the bonding V(B,N) basin. The atomic contribution of the B atom ranges between 0.52 (cetsup) and 0.13e (cofvuo), and is much smaller than that of the nitrogen. The correlation between atomic contribution, B|V(B,N) and N|V(B,N) and the r opt (B,N) has been analyzed with the power regression model, and is presented in Fig. 12. Elongation of the BN bond causes a decrease in the atomic contribution from the N atom, but very small decrease in the atomic contribution from the B atom.
The DI measuring the amount of electron density exchanged between quantum atoms exhibits also some dependence on the r opt (B,N) value. The power regression analysis is

Conclusions
Application of the topological analysis of ELF enabled a detailed description and understanding of the local nature for 37 boron-nitrogen bonds with optimized lengths between 1.241 and 1.785 Å. A total of 25 molecules selected from the CSD have been investigated as well as prototypical H 3 N-BH 3 molecule with a single dative bond (N→B) and the borazine molecule with the delocalized B ... N bond. The BN bonds have been characterized with the wave function approximated within the DFT formalism using the M062X electron density functional in the real and Hilbert space using the topological analysis of η(r) and ρ(r) fields and the NBO analysis.