Structure and Energetics of Complexes of B12N12 with Hydrogen

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Structure and Energetics of Complexes of B N with Hydrogen Halides - SAPT(DFT) and MP2 Study Sirous Yourdkhani, Tatiana Korona, and Nasser L. Hadipour J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b01756 • Publication Date (Web): 14 May 2015 Downloaded from http://pubs.acs.org on May 16, 2015

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The Journal of Physical Chemistry

Structure and Energetics of Complexes of B12N12 with Hydrogen Halides – SAPT(DFT) and MP2 Study Sirous Yourdkhani,∗,†,‡ Tatiana Korona,∗,¶ and Nasser L. Hadipour∗,† Department of Chemistry, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran, Faculty of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Poland, and Faculty of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Poland Tel.: +48-228220211 ext. 525 E-mail: [email protected]; [email protected]; [email protected]

∗ To

whom correspondence should be addressed of Chemistry, Tarbiat Modares University, P.O. Box 14115-175, Tehran, Iran ‡ Faculty of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Poland ¶ Faculty of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Poland Tel.: +48-228220211 ext. 525 † Department

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Abstract Molecular complexes of a fullerene analogue B12 N12 with hydrogen halides (HCl, HBr, and HI) were studied with symmetry-adapted perturbation theory with density-functional theory applied for a description of monomers (SAPT(DFT)), Møller-Plesset theory to the second order (MP2), and its spin-component-scaled variant (SCS-MP2) in a limit of a complete basis set. For each halide five symmetry-distinct minimum structures of the complex have been found on the potential energy hypersurface, with interaction energies ranging from −6 to −18 kJ/mol. The Natural Bond Orbital and the Atom-in-Molecules analysis of noncovalent bonds resulted in a division of these configurations into three categories: hydrogen-bonded, halogen-bonded and those of a mixed type, involving simultaneously a hydrogen bonding and a π −hole bonding between halogen and boron atoms. A comparison of various approaches for the calculation of interaction energies shows that the SCS-MP2 supermolecular method gives results which are in a close agreement with SAPT(DFT), while the MP2 interaction energies are systematically more negative than the SAPT values. The ability of the B12 N12 nanocage to bind hydrogen halides through several active sites on its surface puts under question the selectivity of the binding necessary in crystal engineering, especially for the hydrogen bromide and hydrogen iodide cases, which show small differences in stabilization energies for their minimum structures. The directionality of noncovalent bonds is explained on grounds of the anisotropy of some SAPT components, like electrostatics and induction, as well as by the

σ -hole and π -hole models.

Introduction The emergence of carbon-based fullerenes 1 and nanotubes, 2 and unique properties of these species led to natural questions whether it is possible to devise other types of nanostructures, e.g. constructed from the boron nitride 3–6 (BN) molecule, which is an isoelectronic twin to the C2 unit. Similarly to carbon-based nanomaterials, boron nitrides are promising candidates for many applications, due to their physicochemical properties like: heat stability, high electric insulation charac-


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ter in air, or hydrogen storage ability. For instance, they can serve as electronic devices, high heat resistance semiconductors, insulator lubricants, or nanobioproducts. 7–9 Theoretical studies for finite-size Bn Nn molecules were performed starting from the early 1990s. 10–16 Seifert et al. 12 have studied these species by density-functional theory (DFT) and found that from several Bn Nn nanocages the one with n = 12 is the most stable. Pokropivny et al. 14 predicted (based on experiment and Molecular Dynamics calculations) that even cages as small as B12 N12 or B24 N24 possess the ability to build crystals, what can be used for a construction of molecular sieves and nanomembranes. Other theoretical studies on the BN nanocages, named fulborenes by Pokropivny, have further clarified that the B12 N12 cage of the Th symmetry is more stable than its isomers having the form of graphite-like structures, monocycline rings, or quasispherical cages. 10,11,13,15,16 Finally, about ten years ago the theoretically predicted B12 N12 molecule was synthesized by Oku et al. 17 Among several isomers the one of the Th symmetry showed the largest stability, in a full agreement with the theoretical data. 12 Specific properties of B12 N12 and other nanomaterials from the BN family have led to a surge of demand for extending the areas where these materials can be used. Some of potential applications depend on the strength of the interactions between BN cages and other molecules, therefore the study of physisorption on the B12 N12 surface is an important step towards new potential uses of fulborenes. Noncovalent interactions (NCI), which are responsible for physisorption, can be nowadays studied theoretically using widely available quantum-chemistry programs. In silico investigations allow to obtain many useful data about the NCI, which are otherwise either unavailable or are difficult to extract from experiment (especially if molecules can bind noncovalently in several ways). Among these data one can name details of a geometrical structure and binding energies, where the latter quantity is defined as a difference between optimized electronic energies of the complex (AB) and the constituent molecules A and B (called monomers)

Ebind = EABopt − (EAopt + EBopt ).


(1)


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The stability of the complex can be determined by adding to Ebind the vibrational zero-point energy (ZPVE) difference between the complex and the monomers. Another important quantity of the NCIs is the intermolecular potential energy hypersurface, defined as a difference between the energy of the complex AB and the sum of energies of the monomers A and B

Eint (R) = EAB (R) − (EA + EB ),

(2)

where monomers’ geometries are the same as in the dimer, and where R denotes the set of parameters specifying the positions of A and B with respect to each other. 18,19 The most straightforward way to calculate interaction energies is the so-called supermolecular approach, which consists in a direct application of Eq. (2) with three energies obtained by some approximate method. However, in order to properly characterize the NCIs one should carefully select appropriate tools of computational chemistry from these available “on the market”. An application of methods which are not suitable for intermolecular interactions leads to unreliable results, and even if conclusions of such calculations look sensible, it is the case of “a good answer for a wrong reason”. In particular, for non-polar or electron-rich species the dispersion part of the interaction energy should be evaluated with a sufficient accuracy, what e.g. excludes the simplest Hartree-Fock (HF) theory, which does not account for the Coulomb electron correlation. For the same reason, DFT with many popular functionals cannot be utilized for this purpose, as it does not describe correctly the long-range part of electron correlation. In spite of this common knowledge, there are still many studies of large systems carried out with DFT (some of them have been critically evaluated in Refs. 20,21 for the case of complexes involving fullerenes). As a practical and cheap remedy for this deficiency of the currently available functionals, the DFT-D method has been proposed, where the dispersion correction is evaluated from C6 (and sometimes C8 ) coefficients for atoms and added a posteriori to the DFT electronic energy (see e.g. Refs. 22–24 ). This approach is becoming quite popular nowadays, but it should be noted that its accuracy depends


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heavily on the parametrization and that the so-called dispersion correction can be easily of the same order of magnitude as the DFT interaction energy itself. The Møller-Plesset 25 (MP) method truncated after the second order (MP2), which is another available option to treat large systems, is known for an overestimation of the attraction for complexes involving π -electron systems. 26 Another factor which should be taken into account for each study aiming for accurate energies, is a slow convergence of the interaction energy (especially its part depending on electron correlation) with respect to the orbital basis, what means that sufficiently large basis sets should be used in order to obtain saturated energies. Needless to say, for large molecules this requirement is difficult to fulfill. An alternative way of studying of intermolecular interactions is provided by symmetry-adapted pertubation theory (SAPT), developed by Jeziorski et al., 27–29 which is nowadays a well-established approach available in several program packages. 30–33 A natural perturbation operator in this case is the intermolecular interaction operator, what leaves the sum of the monomers’ Hamiltonians as an unperturbed operator. Opposite to the supermolecular methods, the interaction energy in SAPT is obtained directly as a sum of several energy corrections, and not as a difference of two large and similar numbers, like in Eq. (2). In addition, this method provides us with a decomposition of Eint into energy terms having a clear physical meaning, like electrostatics, induction, and dispersion components and their exchange counterparts, resulting from enforcing of the Pauli exclusion principle on approximate wave functions. This natural partitioning of the interaction energy signifies that SAPT is well suited for studying of the physical nature of the NCIs. Similarly as all other perturbation theories, SAPT would ideally require exact solutions of the unperturbed Hamiltonian, what means energies and wave functions of monomers in this case. Unfortunately, for all but the smallest molecules these solutions are out of reach and they have to be replaced by more approximate wave functions provided by either HF, MP, or coupled cluster (CC) methods, or through the Kohn-Sham (KS) or time-dependent (TD) KS orbitals in the DFT approach. Among these theories the HF one does not account for electron correlation effects inside the monomers, while the utilization of MP and CC is too expensive for many-atom systems.



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This leaves DFT as the only method which can be employed for a description of large interacting monomers. It should be emphasized that while DFT (or TD-DFT for second-order SAPT components) is applied to describe the short-range intramonomer electron correlation, the dispersion effect (the intermonomer electron correlation) is treated entirely through the SAPT model. The SAPT(DFT) method (denoted also as DFT-SAPT) has been developed independently by Misquitta, Szalewicz, Jeziorski et al. 34,35 and Heßelmann and Jansen. 36–39 The comparison of SAPT(DFT) results with both CCSD(T) 40–42 and SAPT(CCSD) 43–47 benchmarks revealed that SAPT(DFT) provides accurate and reliable results, on the condition that a wrong asymptotic behavior of a functional is amended by means of the asymptotic correction. 48 In addition, the implementation of the density-fitting (DF) approach, also known as the resolution-of-identity method, to calculate two-electron repulsion integrals allows to extend applications of SAPT(DFT) to large noncovalent complexes 40,41 (this variant of SAPT will be denoted as DF-DFT-SAPT in the following). Recently, several variants of SAPT, including DF-DFT-SAPT, have been also programmed within the PSI suite of programs 33,49 (see also Ref. 50 ). In the last few years several DF-DFT-SAPT studies of intermolecular interactions of large systems involving fullerenes or molecules of similar sizes have been reported. For instance, Podeszwa has applied this method to study stacked configurations of the coronene dimer and extrapolated them to the graphene dimer limiting case. 51 His results support the claim 52 that the shifted-graphene structure is the global minimum of the coronene dimer. A similar SAPT study has been performed by Totton et al. 53 In another recent work, Jeness and Jordon 54 have applied DF-DFT-SAPT to investigate the interaction of a water molecule with a single graphite sheet by calculating the contributions to the interaction of water with a series of acenes from benzene to dedecabenzocoronene and extrapolating the results to the infinite graphite sheet. One of the present authors (T.K.) has studied the endohedral fullerene complexes with one or two hydrogen molecules 21 and with other small guests. 55 In line with experimental results 56,57 the SAPT(DFT) approach predicted the stability of the complexes: H2 @C60 , H2 @C70 , and 2H2 @C70 , and the instability of the 2H2 @C60 complex. In the same paper 21 a good agreement between


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the spin-component-scaled MP2 (SCS-MP2 58 ) and SAPT(DFT) has been found, while the MP2 method overestimated heavily absolute values of the interaction energies of complexes under study. Unfortunately, this agreement does not persist for all systems, as shown e.g. by Heßelmann and Korona for the case of complexes of C60 with noble gases and for other large species. 59,60 On the other hand, a new hybrid approach MP2C (“coupled” supermolecular MP2), 61,62 which corrects for a deficiency of the MP2 theory by replacing the uncoupled HF dispersion energy by the coupled KS dispersion contribution, provides very similar results to SAPT(DFT) in most of these cases. SAPT(DFT) studies published so far show that this method is a reliable tool to predict the stability of complexes consisting of a fullerene-like nanocage and a second molecule. Therefore, in the present study we applied SAPT for an investigation of noncovalent complexes involving the smallest stable BN-type nanostructure B12 N12 . As the second molecule in the complex we have selected three hydrogen halides HX, where X=Cl, Br, I. From the chemist’ point of view the interaction of HX with B12 N12 is an interesting mix, since both interacting molecules have fragments, which can act as a Lewis base (halogen in HX and nitrogen in B12 N12 ) and a Lewis acid (hydrogen in HX and boron in B12 N12 ). Consequently, it is possible to have halogen and hydrogen bondings, in which B12 N12 acts as a Lewis base. It should be noted that these types of NCIs are of interest in crystal engineering and the formation of the supramolecular architecture. 63–68 The lightest hydrogen halide, i.e. hydrogen fluoride, has been excluded from the present study, since its ability to form a halogen bond is very restricted (it can be detected only if a fluorine atom is linked to a sufficiently strong electron-withdrawing group 69 ). For instance, in a recent study of the interaction of hydrogen halides with borazine the halogen-bonded structure were not found for the fluorine case. 70 To best of our knowledge, the interaction of the B12 N12 nanocage with other molecules has been studied so far on the DFT level only, see e.g. Refs. 71,72 In this contribution we report for the first time the application of more advanced methods, i.e. SAPT(DFT), MP2, and SCS-MP2, to complexes involving the BN-type nanostructures. In addition to the prediction of stability, we use the SAPT-based decomposition of the interaction energy to explore the physical nature of the



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interactions. The final values of the interaction energies are obtained by performing the extrapolation to the complete basis set (CBS) limit, in order to provide saturated values of the stabilization energies. Additionally, we apply the Quantum Theory of Atoms in Molecules (QTAIM) 73 analysis of HF, DFT, and MP2 one-electron reduced densities (1-RDs) for the complexes under study, as well as the Natural Bond Orbital (NBO) method 74,75 in an attempt to connect the parameters of QTAIM and NBO with the stability of various local minima of B12 N12 :HX. We hope that these results will contribute to a better understanding of the character of the BN nanostructures and will stimulate new experiments involving these species.

Computational details Geometries of the B12 N12 :HX (X=Cl,Br,I) complexes were optimized using the M06-2X functional 76 with the aug-cc-pVDZ basis set 77,78 for all atoms but for iodine, for which the aug-ccpVDZ-PP basis set 79 was utilized in order to indirectly account for relativistic effects via incorporating a relativistic pseudopotential. 80 The geometry optimizations were performed with the Gamess program. 81 The analysis of harmonic frequencies at stationary points confirms that these structures are indeed minima. Some test optimizations have been also made with the M06-2X functional in the MG3S 76 basis, and with the MP2 and SCS-MP2 methods and in the def-TZVP 82 (def2-TZVP for iodine 83 ) and aug-cc-pVDZ basis sets. These additional calculations have been performed with the Gaussian09 (DFT) and Turbomole 84 (MP2 and SCS-MP2) programs. The MP2 and SCS-MP2 supermolecular interaction energies were calculated for the optimized dimer geometries (see Eq. (2)). The Boys-Bernardi counterpoise method 85 was applied to reduce the basis-set superposition error, i.e. the monomers’ energies were calculated in the full dimer basis set. All electrons were correlated in these calculations. In addition, we examined the influence of the core-electron correlation on the MP2 interaction energy for B12 N12 :HBr for two choices of frozen bromine orbitals and with the triple-ζ -quality basis sets, which are most frequently used in the production calculations. In our calculations we employed either standard correlation-consistent


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Dunning basis sets and their pseudopotential variants for iodine, i.e. aug-cc-pVXZ 77 and aug-ccpVXZ-PP 86 (X=D,T,Q), or basis sets developed to include the core correlation as well, i.e. augcc-pCVXZ 87 and aug-cc-pwCVXZ-PP 79 sets. The triple-ζ basis was the largest possible set for the complexes with HI and HBr for the core-correlated basis set because of hardware limitations. As an alternative to the supermolecular treatment, the SAPT approach with the DFT description of monomers has been applied. The PBE0 functional 88,89 with the asymptotic correction (AC), as defined by Grüning et al., 48 was utilized as a monomer DFT functional in SAPT. This functional, when used to describe the intramonomer correlation, has been shown to provide reliable intermolecular interaction energies, which are close to the CCSD(T) 40–42 or SAPT(CCSD) benchmarks. 47,90 The ionization potentials, needed for the AC, were taken either from the experimental data (see Refs. 91–93 for HCl, HBr, and HI, respectively), or calculated as a difference of the unrestricted KS energy for the cation and the restricted KS for the neutral molecule for the B12 N12 case, calculated in a large aug-cc-pVQZ basis set with the PBE0 functional. The resulting ionization potential of B12 N12 is equal to 9.77 eV. Additionally, for test purposes, some selected SAPT interaction energies were recalculated with the AC calculated using the theoretical hydrogen halides IPs obtained in the same way as for the fulborene case. The results indicate that the differences resulting from utilizing either experimental or theoretical IPs are small: they amount to at most 0.3 kJ/mol. The details of these calculations are listed in the Supporting Information. All electrons were correlated in SAPT (with one exception discussed in the next chapter). The DF approach for two-electron repulsion integrals was used in Hartree-Fock, MP2, and SAPT(DFT) calculations. For the aug-cc-pVXZ (or aug-cc-pVXZ-PP for iodine) family of orbital basis sets the corresponding JKFIT and MP2FIT auxiliary basis sets 94–96 were utilized. For the aug-cc-pCVXZ (or aug-cc-pwCVXZ-PP for iodine) orbital basis sets, the aug-cc-pVXZ/JKFIT (or aug-cc-pVXZ-PP/JKFIT for iodine) and aug-cc-pCVXZ/MP2FIT (or aug-cc-pwCVXZ-PP/MP2FIT for iodine) auxiliary basis sets 97 were used. The final interaction energies were obtained from the CBS formula for the extrapolation of the electron correlation energy, as proposed by Helgaker et al. 98,99 An alternative extrapolation



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scheme, as proposed by Martin, 100 has been tried as well. The results were found to be similar to the first scheme (usually the differences amount to about 0.2 kJ/mol). For the MP2 and SCSMP2 energies and the aug-cc-pVXZ basis set sequence, the CBS limit was estimated using augcc-pVTZ/aug-cc-pVQZ two-point extrapolation formula, while for the aug-cc-pCVXZ basis sets, the analogous two-basis extrapolation (aug-cc-pCVTZ/aug-cc-pCVQZ) was utilized for the HCl case, while for both HBr and HI complexes the aug-cc-pCVDZ/aug-cc-pCVTZ extrapolation was employed because of lack of aug-cc-pCVQZ interaction energies in these cases. The CBS approach has been applied for DF-DFT-SAPT, too. However, since it is known from many studies 60,101–104 (1)

(2)

(1)

(2)

that the components: Eelst , Eexch , Eind , and Eexch−ind are already well converged in basis sets of (2)

(2)

the aug-cc-pVTZ quality, we applied this procedure to the Edisp and Eexch−disp energies only. The CBS limit has been calculated from the aug-cc-pVDZ/aug-cc-pVTZ two-point extrapolation in this case. The extrapolated dispersion results were then added to the remaining contributions obtained with the aug-cc-pVTZ basis set. The same strategy was applied for the DF-DFT-SAPT calculations using the aug-cc-pCVXZ basis sets. It should be noted that, since the calculation of the coupled exchange-dispersion energy 40 is very time-consuming, 60 it is approximated in this work by scaling the uncoupled exchangedispersion energy by the ratio of the coupled and uncoupled KS dispersion energy. 105 Although this approach has been shown to provide a worse agreement with benchmark SAPT(CCSD) results 90 than CKS terms, the error introduced in this way to the total interaction energy is still small (2)

(2)

because the Eexch−disp component usually constitutes only about 10 to 20% of the Edisp energy at minimum. An alternative approximate method proposed in Ref., 60 which consists in a simple rescaling of the uncoupled contribution by a constant equal to 0.686, can also be used and it gives similar results to the first method. The S2 approximation 106 has been used for all exchange SAPT components. In addition to the interaction energies at minima, the corresponding stabilization energies were


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calculated as

Estab = Eint + Edef + EZPVE .

(3)

In this formula the Eint term denotes the CBS-interpolated SAPT interaction energy and Edef is the sum of monomers’ deformation energies, i.e. differences between electronic energies of the deformed monomers in the complex and these of the isolated monomers. The Edef and EZPVE values used in Eq. (3) were obtained as byproducts of the M06-2X geometry optimization, i.e. the harmonic approximation was applied to calculate the ZPVE energies. All single-point and DF-DFT-SAPT calculations have been performed with the developers’ version of the Molpro suite of codes. 31,32,107,108 The 1-RDs obtained with the M06-2X functional and the aug-cc-pVDZ basis set were analyzed with help of the Natural Bond Orbital (NBO) method, 74,75 as implemented in the NBO 3.1 program 109 within the Gaussian09 package (note that the nondefault occupancy threshold of 1.65 has been used in order to assure that for all complexes and for the isolated fulborene molecule the same types of NBOs are found). Additionally, the QTAIM approach has been utilized for densities produced by Gaussian09 (M06-2X 1-RDs) or Molpro (HF and MP2 1-RDs). The AIMAll program has been used to perform this part of the calculations. 110 For the calculation of the MP2 densities the new Molpro implementation of DF-MP2 analytical gradients 111 has been used, as described in Refs. 112 The molecular electrostatic potential (MEP) on the 0.001 a.u. (e/bohr3 ) contour of the electron density at the M06-2X/aug-cc-pVDZ level was calculated with the WFA-SAS program. 113



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Results and Discussion Geometry optimization The geometry optimization of large intermolecular complexes is a complicated task in view of a multitude of possible local minima. Bearing this in mind, we started the optimization procedure from a careful examination of ways the halogen hydride molecule can approach the fulborene nanocage, basing this analysis primarily on topological properties of the B12 N12 structure. This examination revealed five symmetry-distinct initial configurations, which were then used as starting geometries. No constraints were exercised during optimization. The final structures, depicted in Figure 1, remained quite similar to the initial guesses (with one exception, see below). In particular, the overall positioning of the HX molecules with respect to fulborene persists for all halogens. Therefore, the fact of being a member of the same period of the Mendeleyev table and having the same number of valence electrons plays a dominant role for qualitative features of the complexes under study, while the size of the halogen atom is of a secondary importance in this aspect. This similarity of structures across the halogens allows us to divide the obtained minimum geometries into five types, numbered in the following by Roman numbers. The most important geometry parameters of the optimized structures are listed in Table 1. From the point of view of the predominant interaction type, the structures I–V can be divided into three classes: the halogen-bonded (I), hydrogen-bonded (II and IV), and having four-center (H· · · N–B· · · X–H) hydrogen and π -hole bondings (III and V). The π -bonding has been introduced by Politzer et al. 114,115 some time ago, and it describes a noncovalent interaction arising from the attraction between the region of a positive MEP, which is perpendicular to the extension of the H−X bond, and the electron-rich region of the second molecule. As seen from Figure 2, the depletion of the electron charge is located on the boron atoms of fulborene. For the case of boron the electron deficiency corresponds to an unoccupied pπ -type orbital (i.e. to the 2p orbital of boron contributing to the π bond, see Ref. 115 ). It is quite well documented that geometries obtained on the M06-2X level agree with those


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retrieved from more advanced methods, see e.g. Ref., 116 but usually for these tests large basis sets are utilized. However, in view of the size of complexes under study and number of possible structures we have selected the smallest correlation-consistent Dunning basis with diffuse functions (aug-cc-pVDZ), what in principle can affect the quality of optimization results. Therefore, a question arises to what extent M06-2X/aug-cc-pVDZ minima are universal, i.e. whether similar minima exist on potential energy surfaces obtained by other methods and basis sets. To this end, first the basis set effect has been studied by selecting a triple-ζ quality basis called MG3S, which has been used for noncovalent complexes in the paper of Zhao and Truhlar 76 dealing among others with the prediction of quality of the M06-2X functional. It turns out that if the optimization procedure starts from structures I to V of the complex B12 N12 :HCl, the types I, II, III, and V are reproduced, although in the case III the θNHCl intermolecular angle has changed from 115 to 124 degrees. Unfortunately, we were not able to recover the B12 N12 :HCl(IV) structure with the M062X/MG3S model. Additionally, the complexes with HBr and HI cannot be treated with this basis, since there is no optimized MG3S basis for bromine and iodine. Other optimization tests have been performed with the MP2 and SCS-MP2 methods and the def-TZVP (def2-TZVP for I) basis. The MP2/def-TZVP approach reproduces three, four, and again three structures for complexes with HCl, HBr, and HI, respectively, while SCS-MP2/def-TZVP – two, three, and three structures. Since in some cases SCS-MP2 finds a minimum missed by MP2 and vice versa, altogether four M06-2X/aug-cc-pVDZ structures do not have their counterparts within the MP2 or SCS-MP2 minima with the def-TZVP basis. These configurations comprise the types I and II for B12 N12 :HCl, type II for B12 N12 :HBr, and type IV for B12 N12 :HI. The latter three cases are recovered if the SCSMP2/aug-cc-pVDZ approach is utilized (the MP2/aug-cc-pVDZ model gives the B12 N12 :HI(IV) as well). It should be noted that the differences in intermolecular distances and angles between the MP2 and M06-2X can be significant in some cases, especially for the complexes with hydrogen iodide (e.g. 0.4 Å for the I· · · B distance and 21◦ for the B−I−H angle). However, it should be kept in mind that for the iodine atom different ways of treating core electrons are utilized in both basis sets. The only remaining structure which is not recognized by MP2 or SCS-MP2 approaches,



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is the type I for the complex with hydrogen chloride, which for all studied optimization methods has been found by the M06-2X functional with aug-cc-pVDZ or MG3S basis sets. It is well possible that other tested methods have a saddle point in this region instead of a minimum for the halogen-bonding structure, but in spite of intensive trials we failed to localize this stationary point with either MP2, or SCS-MP2 theories. It seems, however, that the SAPT(DFT) intermolecular energy surface (IES) might have minima for halogen-bonded and hydrogen-bonded structures for the H−Cl−N or Cl−H−N angles very close to 180◦ , respectively, which are close to the M06-2X minima. Unfortunately, a full optimization on the SAPT(DFT) IES is out of reach in view of the lack of the analytic gradients in SAPT, but the cuts performed through the ISE along the Cl· · · N or H· · · N axes, calculated with a larger, aug-cc-pVTZ basis set, show that the respective minimum of the SAPT(DFT) curve and the M06-2X/aug-cc-pVDZ minimum lie very close to each other, and that a modification of the H−Cl−N or Cl−H−N angles by 10 degrees while keeping the distance between the center of mass of HCl and the closest N atom constant, leads a decrease of absolute value of the interaction energy. Summarizing, the SAPT(DFT)/aug-cc-pVTZ results support the existence of typical Cl-bonded (type I) and H-bonded (type II) structures, what makes us to keep all M06-2X/aug-cc-pVDZ minima in our analysis. The detailed optimization results for all utilized models and the abovementioned SAPT(DFT) curves can be found in the Supporting Information.

Structural features of the complexes Before we start to analyze the minimum geometries in detail, it is worth to notice (see Figure 1) that no structure shows a direct interaction between the hydrogen atom of HX with the boron atom of the cage. This fact can be explained by the small electronegativity of both H and B atoms, what makes them electron-deficient in their respective monomers. Therefore, the observed bondings between B12 N12 and HX are always created between the electron-deficient and electronrich partners, in a full agreement with chemical intuition. The interatomic distance presented in Table 1 are 0.1 up to 0.3 Å smaller than a sum of van der Waals radii, 117 what suggests the existence of a noncovalent bonding. The length of the H-X bond (see ∆RHX in the table) is always larger in 14 ACS Paragon Plus Environment

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the complex than in the isolated hydrogen halide, but for most cases this increase is very small, indicating a weakness of the intermolecular bonding. In the third and fifth columns of Table 1 the distances between the closest atoms from both monomers are presented. The examination of this data reveals that one should expect the smallest stabilization energy for the type I complexes, as their interatomic distances are the largest from all five structures, while for the type II the largest stabilization is expected, because the intermonomer distances are the smallest in this case. A perusal into Table 2, which collects the final results for the interaction and stabilization energies, confirms these conclusions. In the configurations I and II the hydrogen halide molecule is placed on the top of the NB3 unit of fulborene. These linear structures resemble a much simpler system consisting of the ammonia molecule interacting with HX, but the geometry optimization of the H3 N· · · XH (type I) and H3 N· · · HX (type II) complexes, obtained with the same (M06-2X/aug-cc-pVDZ) optimization method, reveals that nitrogen in ammonia has a weaker effect on the length of the HX bond if the hydrogen halide approaches ammonia from its halogen side, while the opposite is true for the hydrogen bonding. In the latter case the stretching of the H-X bond and shrinking of the X-N distance are so pronounced that the hydrogen atom is actually shared between halogen and nitrogen. Therefore, the boron atoms around nitrogens in fulborene play a double role by enhancing the ability of the halogen-bond formation and by disfavoring the creation of hydrogen bonds. Let us move to a more detailed examination of the optimized structures. As expected, for the type I complexes (see Figure 1a), a distance between the halogen X atom and the B12 N12 molecule increases with the atomic number of X. The HX molecule points its halogen ending to the nitrogen atom and the H-X-N angle is very close to 180◦ . Such directionality, which is a characteristic feature of both hydrogen and the halogen bondings, makes this type of interactions potentially interesting for crystal engineering. 115,118 However, in the case of the complexes with B12 N12 this structure is the least stable from all five possibilities, so it can be expected that other, more stable minima will be selected during complexation in the first place. In the B12 N12 :HX(II) structures (see Figure 1b) the HX molecule is linked to the nanocage



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through the N· · · H contact. A large difference between the N· · · H distances and the sum of the Van der Waals radii and the closeness of the N-H-X angle to 180◦ confirm that this configuration is a hydrogen-bonded complex. The N· · · H distance decreases in the series I>Br>Cl, what indicates that the smallest and the most compact chlorine atom provides us with the shortest and the strongest hydrogen bond. A strong anisotropy of the both halogen and hydrogen bondings is usually explained through the σ -hole concept, 114,119 which in the case of hydrogen halides denotes positive regions of the MEP around the H-X bond axis at its hydrogen or halogen endings. The directionality of these bonding types can be also explained by the analysis of interaction energy components, see Ref. 120 and the next section. A characteristic feature of both type III and V structures (see Figure 1c and Figure 1d, respectively) is a position of the HX molecule, which lies approximately parallel to one B−N bond of B12 N12 . The difference between these two geometries results from two topological types of these bonds which are present in the nanocage under study: for the case of the B12 N12 :HX(III) complexes, the HX molecule lies above the B−N bond shared by a tetragonal and a hexagonal ring, while for the B12 N12 :HX(V) case the hydrogen halide is placed above the B−N bond shared by two hexagonal rings. For both configurations the halogen ending of the hydrogen hydride is oriented towards the boron atom, while the hydrogen ending points towards the nitrogen atom of the B12 N12 molecule. In both cases the N· · · HX distances are substantially smaller than the X· · · B ones. All geometrical parameters are very similar for the case of the X· · · B bond. The only significant difference of 0.2 Å has been found for the lightest halogen. These optimal orientations are in a good agreement with the electron charge distribution around the hydrogen and halogen atoms (see Figure 2). A negative MEP belt 69 around the latheral side of HX gives the Cl atom the ability to interact (noncovalently) with electropositive sites for angles of about 90◦ with respect to the H–X axis. The occurence of this belt can be explained by the presence of electrons on both np orbitals of halogens (n = 3, 4, 5 for Cl, Br, I, respectively) perpendicular to the H−X bond axis. Finally, for the type IV minima, the HCl and HBr molecules act as hydrogen donors, what


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allows to classify this case as an suboptimal version of the hydrogen bonding (see Figure 1e). A similar structure could not be reproduced for the HI molecule in spite of intensive trials: all optimization attempts lead to a structure with the HI molecule attached to the B12 N12 cage through the I· · · B contact (see Figure 1f). The N· · · H distances in the case of Cl and Br and the B· · · I distance are smaller than the sum of the Van der Waals radii (a decrease about 0.62, 0.42, and 0.23 Å is observed, respectively), what indicates the presence of the noncovalent bonding. The observed angles between the HX and the nearest atom of B12 N12 are about 82◦ . The B· · · I distance is equal to 3.40 Å, what is less than analogous distances for the B12 N12 :HI(III) and B12 N12 :HI(V) configurations. This indicates a stronger bonding for the present type (IV) than for the partial halogen-type bonding represented by the types III and V. Unusual geometry preferences of the HI molecule on the B12 N12 surface can be explained by the existence of a negative belt in the MEP for iodine, which is more extended than for the ligher halogens, as shown in blue in Figure 2. Apparently for the case of HI, the interaction resulting from this negative belt prevails over the interaction with the hydrogen atom, i.e. HI behaves as a classical Lewis base. These special features of iodine were already reported in the literature. 121

Energies of the complexes The CBS-extrapolated interaction energies from the SAPT(DFT), MP2 and SCS-MP2 methods are listed in Table 2. In the next two tables (Table 3 and Table 4) the best SAPT(DFT) components are presented. Unfortunately, we were not able to obtain all-electron DF-DFT-SAPT results in the quadruple-ζ basis sets, as well as DF-MP2 results in the aug-cc-pCVQZ and aug-cc-pCVTZ basis sets for the B12 N12 :HBr and B12 N12 :HI complexes. For this reason we utilized the augcc-pVXZ hierarchy as our primary basis sets, while incomplete aug-cc-pCVXZ results were used to estimate the importance of the correlation with core electrons for the intermolecular interaction energy. In particular, a comparison of the aug-cc-pCVDZ/aug-cc-pCVTZ and aug-cc-pCVTZ/augcc-pCVQZ extrapolation limits in the chlorine case and the aug-cc-pVDZ/aug-cc-pVTZ and augcc-pVTZ/aug-cc-pVQZ extrapolation for all halogens for the MP2 and SCS-MP2 energies allows 17 ACS Paragon Plus Environment


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to assess the error of the extrapolation in the double-to-triple ζ sequence to about 0.05 kJ/mol. We have verified that the larger uncertainty comes from the usage of the alternative CBS-extrapolation procedure proposed by Martin, 100 which gives interaction energies about 0.1 up to 0.2 kJ/mol higher than the method of Helgaker et al. 98,99

Energetic order of B12 N12 :HX structures In the last column of Table 2 the stabilization energies calculated according to Eq. (3) are presented for all complexes. As expected, these energies are less negative than the interaction energies, and in one case (the halogen-bonding case for B12 N12 :HCl) a small instability of the complex is predicted. The energetic order of stabilization and interaction energies is the same for the chlorine and the bromine case. A different situation occurs for iodine, where all interaction energies are very similar to each other and small differences in the deformation and zero-point vibrational energies are enough to change the energetic order of the structures (the type III complex becomes the most stable). It turns out that the major part of the destabilization effect comes from the EZPVE contribution, while the geometries of the B12 N12 nanocage and the HX molecules change only a little under interaction with each other, what for the case of hydrogen halides is documented in Table 1. Small differences between the stabilization energies lead to the conclusion that all studied minima are energetically accessible at room temperature (and even at much lower temperatures), what means that the selectivity of the complex formation from HX and B12 N12 will be low at the gas phase. We should remember, however, that the B12 N12 molecules form crystals and other regular structures, 122 so only selected face and vertex types of B12 N12 remain accessible on the crystal surface, what in turn excludes some types of the B12 N12 :HX configurations. Let us now take a closer look at the energetic aspects of the structures. As it can be seen from Table 2, the MP2 method overestimates the complex binding, while SCS-MP2 usually gives smaller absolute values of the interaction energies in comparison to SAPT(DFT). Interestingly, the SCS-MP2 interaction energies obtained with core-valence basis sets show a better agreement with the corresponding SAPT(DFT) than for the case of the valence basis sets. In each case the largest


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difference between these two methods occurs for the hydrogen-bonded structures (type II). The basis-saturated interaction energies predict the same energetic order of the local minima regardless the electron-correlated method used for computations. For the B12 N12 :HCl complex the absolute value of Eint becomes smaller in the II>IV>V≈III>I sequence. Different configuration types turn to be energetically closer for heavier halogens: for the B12 N12 :HBr complex this sequence becomes II>III≈V≈IV>I, and for the iodine-containing complex only one structure is significantly less stable than the others, so the order becomes I≈II≈III≈V>IV. The equalization of the energy of most structures comes in line with narrowing the energy span between the most and the least favorable structure, which amounts to 13 kJ/mol, 8 kJ/mol, and only 1.5 kJ/mol for B12 N12 :HCl, B12 N12 :HBr, and B12 N12 :HI, respectively. The B12 N12 :HCl structures show not only the largest energy differences, but also contain the smallest and the largest stabilization examples among all three halogens. The B12 N12 :HCl(II) minimum is the most stable with the SAPT interaction energy equal to −18.09 kJ/mol, while on the opposite end the type I minimum is the weakest one with the SAPT interaction energy equal to −5.58 kJ/mol only, what means that the latter structure is almost three times less favorable energetically than the former one. An analysis of the bonding type (see Figure 1) reveals that the highest stabilization occurs for the hydrogen-bonded complex, i.e. with the hydrogen atom from HCl pointing towards one of the nitrogen atoms of the cage, while in the weakest minimum two monomers are linked through the N· · ·Cl halogen bond. Summarizing, the hydrogen chloride molecule clearly prefers to act as a hydrogen-bonding acceptor instead to bind through its halogen ending. This tendency can be also confirmed for the heavier bromine (although in this case the energy ratio between the interaction energies of the II and I structure types is smaller than in the chlorine case), and it dissapears completely for the heaviest halogen under study, where the structures I and II have essentially the same energy. For the halogen-bonded case the order of stability corresponds to the halogen radius decrease, i.e. one has B12 N12 :HI(I)>B12 N12 :HBr(I)>B12 N12 :HCl(I). This sequence shows that as the radius of the halogen atom grows, its ability to form a stronger halogen bonding becomes larger, too,



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with the interaction energies increasing nearly twice between the lightest chlorine and the heaviest iodine. These differences can be also explained through the σ -hole model 69 (see the discussion of geometries). The σ hole, which is a positive MEP region in the extension of the H−X bond, is depicted in Figure 2b, Figure 2c, and Figure 2d for X=Cl, Br, and I, respectively. The figures reveal a much more positive σ -hole for the iodine atom in comparison to the lighter halogens, what explains its ability to bind electron-donating species. This trend comes in line with weakening of hydrogen-bonded complexes created from heavier hydrogen halides. Let us turn the attention to the type IV structures. As already mentioned during the geometries discussion, we put under this name the suboptimal (i.e. nonlinear) H-bonded complexes for chlorine and bromine and a structure for iodine, which binds to the B12 N12 cage through the boron atom. Such a selection of the attachment place can be justified by a presence of the negative MEP belt for the halogen bond. The deviation from linearity for the case of the hydrogen-bonded B12 N12 :HCl(IV) and B12 N12 :HBr(IV) complexes results in a stability loss as compared to the corresponding type II cases. Quite on the contrary, the interaction of the hydrogen iodide molecule through the negative MEP belt of iodine results in a complex, which is energetically as strong as the hydrogen-bonded structure. Two mixed four-center configuration (types III and V) show a similar stability for the same halogen, so one can conclude that there is energetically no difference between the interaction with two topologically distinct B−N bonds of B12 N12 . Energy decomposition and importance of core correlation Let us now focus on the components of the SAPT(DFT) method in order to gain insight into the physical origin of the interactions. The SAPT results presented in Table 3 and Table 4 show the SAPT(DFT) components at the CBS limit, for the aug-cc-PVXZ and aug-cc-pCVXZ basis sequences, respectively (the components of SAPT for individual basis sets can be found in Supporting Information). Evidently, the core-valence basis would be preferable for all-electron correlated calculations, but since we were not able to perform DF-DFT-SAPT calculations for the triple-


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ζ -quality basis for B12 N12 with both heavier hydrogen halides because of hardware limitations, we have chosen to utilize the aug-cc-pVXZ CBS-extrapolated energies for consistency of all results. We can use, however, the available results obtained with the second basis hierarchy to gauge an importance of the core-electron contribution to various components of the interaction energy, since in the aug-cc-pCVXZ basis sets some extra functions for a description of core correlation are present, 123 while the absence of these functions in the aug-cc-pVXZ basis can artifically shrink core-valence contributions to the interaction energy. As can be seen from Table 3 and Table 4, the SAPT components resulting from both sets differ by as much as 1 kJ/mol for the HCl-containing complexes, while they are almost identical for HBr. In the latter case small differences in various SAPT terms add to at most 0.05 kJ/mol value for the total interaction energy, what is even less than the differences between SCS-MP2 and MP2 values for both basis sets. Although the main results of this paper are obtained with the explicit treatment of electron correlation for all electrons (with the exception of iodine, where the ECP is used), we additionally performed several calculations, which aim to estimate the magnitude of the core correlation on the intermolecular interaction energies. The examination of this effect on MP2 interaction energies is particularly straightforward, since in this case it is enough to exclude from all-electron results those pair MP2 energies contributions, which contain an excitation from at least one frozen orbital. We tested two frozen-orbital schemes for the case of the the B12 N12 :HBr complex for both triple-ζ basis sets. The first frozen-core set consists of 1s orbitals of boron and nitrogen and all orbitals from the K, L, and M shells for bromine, while in the second scheme the 3d orbitals are left unfrozen. The results presented in Table 5 show that freezing of the 3d subshell of bromine has a sizable effect on the interaction energies (errors of order of 3% with respect to the all-electron calculations have been found for aug-cc-pVTZ, while this error is, on average, as large as 4.5% for the aug-cc-pCVTZ basis). On the other hand, freezing the 3s and 3p bromine orbitals has a much smaller effect on Eint . It appears therefore that the 3d orbitals of Br contribute to some extent to the noncovalent binding. A comparison of orbital contours of the highest occupied and lowest unoccupied orbitals of B12 N12 and 3d Br orbitals reveals that they indeed have the same sign pat-



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tern and can therefore effectively overlap, especially that the 3d orbitals have much higher orbital energy than the 3p or 3s bromine orbitals. The B12 N12 :HCl complex has been used for examining of the core-electrons’ effect on SAPT(DFT) components. The 1s orbitals for B and N and 1s2s2p orbitals of Cl were frozen in these computations and the aug-cc-pVDZ/aug-cc-pVTZ CBS extrapolation was utilized for the dispersion and exchange-dispersion components, exactly as it was done in Table 3. The final CBS-extrapolated results are listed in parenthesis in Table 4 below the corresponding aug-cc-pCVXZ values. A comparison of both SAPT components reveals that they are very close to each other. As a matter of fact, the frozen-core SAPT results in the valence basis agree better with the all-electron core-valence components than with the all-electron valence-basis terms from Table 3. If this observation persists for other cases, it will signify that in some cases it is safer to use either frozen-core SAPT with the valence-type basis sets or all-electron SAPT with the core-valence basis, than the all-electron SAPT with the valence-type basis. Such results can be explained by the basis-function "borrowing" for needs of the core-valence correlation, what results in a deterioration of the description of the (main) valence-valence correlation. A further examination of Table 3 and Table 4 allows to find possible coincidences between the type of intermolecular bonding and the importance of various SAPT terms. First let us make some general observations concerning SAPT components. For all complexes the electrostatic con(1)

tribution is negative, but the first-order exchange term is several times larger than Eelst , so the total contribution from the first-order SAPT is strongly repulsive. Therefore, the stability of the structures is mostly due to the second-order contributions, among which the dispersion energy plays (2)

a dominant role, because it is never quenched by its exchange counterpart as much as the Eind (2)

(2)

(2)

component (the Eexch−disp term constitutes usually about 10% of |Edisp |, while Eexch−ind is often of (2)

order of 90 to 99% of |Eind |). The dispersion term grows from HCl to HBr, what is consistent with an increasing number of electrons, but it is similar for HBr and HI, what can be explained by the fact that not all electrons in iodine are treated explicitly and electrons from the ECP cannot contribute to the dispersion energy. The selected higher-order contributions are usually approximated


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by the so-called δ Hartree-Fock term (δ EHF ), which is calculated as a difference between the supermolecular Hartree-Fock interaction energy and the sum of electrostatic, first-order exchange, second-order induction and exchange-induction terms with monomers treated on the uncorrelated (i.e. Hartree-Fock) level. 124,125 The usage of this correction is strongly recommended in presence of the ECPs for some atoms. 126 Since in the DF-DFT-SAPT code utilized by us in the present study the first-order exchange term is treated within the S2 approximation, an additional gain from (1)

the δ EHF term is an indirect inclusion of higher-than-single exchange contributions of Eexch . It should be noted that in an ideal SAPT application this term should be small in comparison to other components, as it is supposed to contain mostly third and higher-order induction and exchangeinduction contributions. Results listed in Table 3 and Table 4 show that it is unfortunately not the case: the δ EHF term constitutes an important part of the interaction energy, especially for the structures of type II and IV. In such cases one can expect a somewhat poorer agreement of SAPT with benchmarks, but nonetheless the inclusion of δ EHF is always beneficial for the accuracy of the interaction energy. 127 This term the most negative for the hydrogen-bonded structures, and the least negative for the halogen-bonded ones. Actually, for the B12 N12 :HI(I) configuration the δ EHF becomes positive (it is interesting to note that a difference between corrections for the type I and type II structures remains similar – from 7 to 9 kJ/mol – for all three pairs of complexes). Let us start a more detailed examination of the tables from the weakest B12 N12 :HCl(I) structure. (2)

In this case the largest attractive contribution to the ESAPT is the Edisp term, while the netto effect of int the induction and exchange-induction is close to zero. The electrostatic contribution for this struc(2)

ture is the smallest one (in absolute value) for all complexes under study. The Eind (B12 N12 → HX) components are equal to −0.57, −1.25, and −2.22 kJ/mol for the B12 N12 :HCl(I), B12 N12 :HBr(I), (2)

and B12 N12 :HI(I) complexes, respectively, while the Eind (HX → B12 N12 ) components amount to −6.04, −15.53, and −24.55 kJ/mol. These results show that the fulborene cage is easily polarized by hydrogen halides, while the polarization in the opposite direction is much less pronounced. This asymmetry of the induction effect can be easily explained by the common picture of a multipole inducing a dipole on the second monomer (the B12 N12 is highly symmetric, so its long-range



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electric field is much weaker than for the HX case), but the magnitude of polarization itself cannot (2)

be elucidated by this simple model, according to which the largest Eind (HX → B12 N12 ) component should be expected for X=Cl (since the HCl molecule has the largest dipole moment among all three hydrogen halides under study). In order to explain the increasing absolute values of the induction energies one should remember that in the Van der Waals minimum region the SAPT components result from the overlap of the unperturbed and perturbed electron clouds of the respective monomers. It is only when the distance between monomers becomes sufficiently large, the multipole interpretation of the interaction can be safely utilized. It can also be seen that with a growing atomic number of halogen the induction effect becomes (2)

more and more important. For the B12 N12 :HBr(I) structure the Eind term already becomes the largest attractive individual correction, but analogously as in the chlorine case, it is almost completely quenched by its exchange counterpart. A different situation occurs for the B12 N12 :HI(I) (2)

(2)

structure, where only about 75% of the Eind is cancelled by the Eexch−ind term. Therefore, in contrast to the B12 N12 :HCl(I) and B12 N12 :HBr(I) complexes, which are effectively stabilized by the dispersion energy, in the B12 N12 :HI(I) complex the induction energy also plays a sizable role in the stabilization effect. In the two former cases, the δ EHF correction contributes about 18% to the total interaction energy, while in the latter case its role grows to as much as 45%. It should be mentioned that in the B12 N12 :HI(I) stabilization energy, the δ EHF term plays a significant destabilization role in a contrast to the two former cases. This discrepancy can be attributed to the ECP model for the iodine which prevents the core electrons from being involved into polarization. As already noticed, such large contributions of this correction to the total interaction energy can signify that higher order SAPT should be taken into account, and that without higher order terms the SAPT results might be less reliable. 127 As already mentioned, on the other end of the interaction scale there is another chlorinecontaining structure of a typical hydrogen-bonding features, like a 180◦ N–H–Cl angle, elongated H–Cl bond, and the smallest HCl· · · B12 N12 distance among all structures. As usual for such a bonding, the largest attractive SAPT component is the electrostatic energy. In the case


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of B12 N12 :HCl(II) the electrostatics is about four times more negative than the halogen-bonding (1)

case. Also for B12 N12 :HBr(II) the Eelst term is the largest in the absolute value among all contributions, but for the B12 N12 :HI(II) case the electrostatics, although still large, is shifted from the (2)

first place by the dispersion term. The dominating role of Edisp in the latter case can be explained by the large number of electrons in iodine (even after taking into account that not all electrons in (1)

iodine are treated explicitly). Although the Eelst term is the major attractive contribution in the B12 N12 :HCl(II) and B12 N12 :HBr(II) complexes, the total contribution from the first order of SAPT is repulsive because of the large first-order exchange term. Contrary to the halogen-bonding complexes, a large (in absolute value) second-order induction component is quenched by its exchange component by about 65% on average only, what means that the netto induction part plays an important stabilizing role in the complex. Additionally, the structures are stabilized by the dispersion energy, which has a similar magnitude as induction. Therefore, a popular interpretation of the hydrogen bonding as a result of the interaction of a static electron cloud of the Lewis base with the hydrogen core should be actually corrected by allowing for the relaxation, or polarization of these clouds under influence of another monomer. Summarizing, it can be concluded that although the electrostatic contribution plays some role in the stabilization of the B12 N12 :HX(II) complexes, the induction and dispersion energies are indispensable to arrive at the negative interaction energy. Since the B12 N12 :HX(IV) (X=Cl,Br) structures can be regarded as the corresponding bent configurations for the B12 N12 :HX(II) (X=Cl,Br) counterparts, it is interesting to analyze the influence of bending on the SAPT components. According to Table 3 and Table 4, all first-order and secondorder induction and exchange-induction terms become smaller (in absolute value) compared to the corresponding terms for the B12 N12 :HCl(II) structure. Only the dispersion term remains practically the same (for the chlorine case) or even becomes 1.4 kJ/mol more negative (for the bromine case). This finding is in line with the character of dispersion, which is usually the least anisotropic among the low-order SAPT contribution. The same pattern of the SAPT components has been found for the B12 N12 :HI(IV) structure (which is however of a different bonding type), except that this time the dispersion energy becomes lower by almost 4 kJ/mol.



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In the structures of type III and V the hydrogen halide molecules interact simultaneously via the their hydrogen side and the negative belt of the halogen with the nitrogen and boron atoms of B12 N12 . Because of the partial hydrogen bonding we expect that the induction energy will not be completely quenched by the exchange-induction term, as for the pure halogen-bonded structure. The inspection of the tables shows that this is indeed the case: the netto part of the induction contribution amounts to about –3 to –4 kJ/mol. A comparison of the SAPT components for the types III and V structures reveals that they are quite similar to each other. The differences between corresponding components are not larger than 1.5 kJ/mol and can be attributed to geometrical differences between the closest environment of the B−N bonds directly involved into the interaction with HX. Summarizing, the SAPT approach shows that the interactions of the hydrogen halide molecule and of tetragon-hexagon or hexagon-hexagon edge of the nanocage are very similar to each other. The analysis of the interaction energy components allows to present some explanations to discrepancies between SAPT(DFT) and MP2 interaction energies. The dispersion energy plays a dominating role in the attractive part of the interaction energy for all studied structures (with some contribution from induction for hydrogen-bonded cases), so its description on the intramonomer correlated level is of utmost importance. In SAPT(DFT) it is evaluated with the use of CKS propagators. On the other hand, the MP2 interaction energy includes the dispersion energy on the simplest, uncoupled HF level, 128 which usually leads to an overestimation (in absolute value) of the dispersion component. This effect is especially important for complexes with monomers containing conjugate bonds, 26 where intramonomer correlation corrections to the dispersion are expected to play a nonnegligible role because of a large number of electrons squeezed together in double bonds.

Atoms In Molecules The quantum theory of atoms in molecules (QTAIM) 73,129 is a useful tool for visualizing chemical interactions, including non-covalent ones, such as hydrogen or halogen bondings. In the QTAIM 26 ACS Paragon Plus Environment

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a topological analysis of the one-electron reduced density ρ (called also the charge density in the following) is performed by calculating its gradient vector field. 130 By inspecting of the density distribution between two adjacent atoms, a line for which the density has a maximum value in two directions, can be identified in some cases. This line of the maximum density is sometimes called an atomic interaction line or – more often – a bond path (BP) 131 (note that this second name caused a lot of controversies and misunderstandings, see Refs. 132–135 ). The point of the BP where the value of ∇ρ is zero is called a bond critical point (BCP). At the BCP one can calculate quantities such as the charge density (ρBCP ), Laplacian of ρ (∇2ρBCP ), the total electron energy density (HBCP ) and its two components, i.e. the kinetic electron energy density (GBCP ) and the potential electron energy density (VBCP ), which provide a useful information about the strength of the interaction between two atoms 73 and which are also used for an interaction classification. 136,137 Remembering that the interactions in molecules are of electrostatic nature, it is still useful to divide them into two main types: the so-called open-shell interactions (or shared-valence ones), which lead to strong, or chemical bonds, and closed-shell interactions, which represent much weaker interactions between closed-shell subsystems. Energetic properties of the charge density are related to its Laplacian at the BCP by identities, 73 1/4∇2 ρBCP = 2GBCP + VBCP

and HBCP = GBCP + VBCP .

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In the framework of the QTAIM, the negative sign of both ∇2 ρBCP and HBCP indicates an openshell interaction between two atoms, while ∇2ρBCP >0 and HBCP > 0 are the indicators of a closedshell interaction. If ∇2 ρBCP >0 and HBCP < 0, the interaction is partly covalent, i.e. is an intermediate between the open-shell and closed-shell case. 73,129,137–140 These typical features of open- and closed-shell interactions can be understood from the analysis of the density ρ and its derivative quantities at the BCP. For instance, if the value of ∇2ρBCP is less than zero at a given point, the density is locally concentrated, whereas ∇2ρBCP >0 indicates a local depletion of ρ . Therefore, the negative value of ∇2ρBCP indicates the occurence of a cova-



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lent bond, in which a sharing of electrons (or – more precisely – of the charge density) happens, while its positive value indicates a closed-shell interaction. Based on Eq. (4) one can conclude that ∇2 ρBCP can be interpreted in terms of a balance between the local kinetic and the local potential electron density. 141 Therefore, the ratio of −GBCP /VBCP also provides some information about the character of the interactions. To this end, if −GBCP /VBCP is greater than 1, the interaction is a closed-shell one, in the case of the −GBCP /VBCP value ranging between 0.5 and 1 the interaction is partly covalent, and if the ratio of less than 0.5, the interaction has as a shared-covalent nature.

Closed-shell nature of interaction of B12 N12 with HX Properties of the charge density at BCPs for the B12 N12 :HX complexes are listed in Table 6 for the intermolecular BCPs for the MP2 densities obtained from the aug-cc-pVDZ basis. The same analysis for the M06-2X and HF densities can be found in the Supporting Information. Let us start with the particular properties of the minima types I to V. Among the B12 N12 :HX(I) complexes the values of charge density are equal to 0.0104, 0.0092, 0.0081 for N· · · I, N· · · Br, and N· · · Cl, respectively, i.e. they decrease with the atomic number of the halogen atom. This trend is in a good agreement with the energetic (GBCP and HBCP ) results from the same table, which show that the strength of halogen bondings decreases in the same order. For all the complexes the value of −GBCP /VBCP is larger than one and the Laplacian of ρ is positive, what clearly identifies the N· · · X interactions are of a closed-shell nature. For the second orientation of complexes, i.e. B12 N12 :HX(II), the charge densities at BCP for the H· · · N BP are equal to 0.0235, 0.0225, and 0.0166 a.u., in the trio of atoms: Cl−H· · · N, Br−H· · · N, and I−H· · · N, respectively, i.e. the largest charge density occurs for the lightest halogen atom, what agrees with the energetic order given in Table 2. The value of ρ is about two times larger than for the halogen bonding discussed above, what supports the observation that the hydrogen bond is the strongest among the non-valence bonds. As can be seen from Table 6, also in this case the sign of the Laplacian of ρ shows that the electron charge is depleted, as usual for the closed-shell bondings. It should be also noted that the ellipticity (defined as λ1 /λ2 − 1) is equal


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to zero for both structure types I and II, what agrees with the linearity of these bond types (see Table 1). The situation becomes more complicated for the case IV, where the hydrogen iodide plays a different role than its lighter analogues, namely, in contrast to HCl or HBr, which interact with a nitrogen atom from the B12 N12 cage, the HI molecule forms a bonding with a boron atom. For the two lighter atoms, the values of the charge density at BCP of N· · · H−Cl and N· · · H−Br are 0.0198 and 0.0141 a.u., respectively, i.e., they become smaller as the halogen atom becomes heavier. It can be therefore anticipated that for the heaviest iodine atom the interaction N· · · H−I is too weak and therefore it is apparently overrun by a stronger interaction with boron. These changes indicate the effect of the value of the θNHX angle on the strength of such interactions. Other features from the tables, i.e. the Laplacian at BCP of I· · · B in B12 N12 :HI(IV) complex, HBCP , −GBCP /VBCP > 1, allow to classify the I· · · B interaction as the closed-shell type. The B12 N12 :HX(III) and (V) complexes turned out to be the most problematic for the QTAIM. In both cases the analysis of the position of HXs with respect to the B12 N12 cage, which is more or less parallel to one B−N bond, as well as the comparison of the atom-atom distance with the sum of Van der Waals radii (see Table 1) let us naively anticipate that the BPs corresponding to both interactions: halogen-boron and hydrogen-nitrogen should be present. However, the results from Table 6 show that the expected boron to halogen BP has been detected for the B12 N12 :HBr(III) complex only, but for this case no hydrogen to nitrogen BP has been found. Three H· · · N BPs out of expected six are present in the table, and instead of expected BPs we are often confronted with "unusual" ones between the nitrogen and halogen atoms, which are both Lewis bases and as such should not create a bond. 69 The absence of some expected BPs can be understood only after the analysis of the whole charge density. The contour map shown in Figure 3 reveals there is an almost flat region within the B-N-H-X rectangle (in agreement with the smallness of λ2 from Table 6), what explains why the QTAIM detects a whole variety of BPs depending on the complex and the method and does not find the second BP connected to the boron atom. It turns out that if distances between B12 N12 and HI become smaller, both BPs (H· · · N and B· · · I) reappear, but the



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elongation of this distance causes the B· · · I line to become more curved towards the B−N bond (see the Supporting Information). Another possibility to analyze the interaction character, which will be exploited here, is based on the topological properties of the Laplacian of the charge density. This quantity tells us where the charge density is locally concentrated or locally depleted. A relief map of the ∇2ρ for the B12 N12 :HCl(III) complex (see Figure 4) shows indeed that the charge is locally concentrated around chlorine and depleted in a valence region of the boron atom. Therefore, from a static point of view, there should be an electrostatic attraction between halogen and boron. This indirect analysis, together with the B−Cl distance, which is smaller than the sum of Van der Waals radii of B and Cl, can be viewed as a confirmation of a noncovalent bond between these two atoms. Summarizing, the QTAIM analysis of the BCPs confirms that the intermolecular BPs of the closed-shell nature, but it does not detect some expected atom-atom connections. An explanation of this fact has been proposed based on the contour maps of the charge density. Our analysis of the problematic cases for the III and V structures fits into the already reported problems with the simplistic interpretations of the properties of intermolecular BCPs . 142–146

Delocalization indices Yet another possibility to study the intermolecular interactions is to perform the analysis of the delocalization index (DI), which is obtained from an integration of the two-electron charge density over the basins of investigated atoms. 147–149 The DI, denoted as δ (ΩA , ΩB ), provides a measure of the extent to which the electrons are delocalized between two atoms. The values of DI computed at the HF and MP2 levels are listed in Table 7. First of all, we can see that the sizable DI values coincide with the detected BCPs for the complexes of type I, II, and IV. The DI analysis recognizes a different character of the hydrogen iodide (IV) complex: while for the HCl and HBr cases the

δ (N, H) is different from zero, for the HI case the δ (I, N) and δ (I, B) are nonzero instead. For III and V types of minima, which cause so many interpretation problems, all three indices: δ (X, N),

δ (X, B), and δ (N, H), are simultaneously nonzero. Therefore the DI analysis shows that we have


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to do with a concerted electrons’ displacement between four atoms, what can partially explain a failure of finding some features of such interaction through the analysis of local properties of 1RDs. It is interesting to notice that MP2 values of the δ (X, B) index are almost twice as large as the HF ones, what can reflect both the importance of electron-correlation and differences in a definition of electron-correlated and HF DIs. 149 A multicenter bond index, which is a generalization of the DI applicable to delocalized n-center bonds and rings has been proposed some time ago by Giambiagi et al. 150 Very recently, this index has been utilized for studying cooperative effects in hydrogen bonds. 151 In the latter work, the following definition of the n-center index has been propoesd for the Hartree-Fock case,

δ (A1 , A2 , . . . , An ) =

nocc nocc

∑ ∑

i1 =1 i2 =1

nocc

...

∑ Si1i2 (A1)Si2i3 (A2) . . . Sini1 (An),

(5)

in =1

where Si j (A) is the overlap integral of occupied orbitals i and j over the basin of the atom A. In order to extend our analysis, we calculated 3-center and 4-center indices from Eq. (5) for atoms possibly involved into the interaction (i.e. both atoms of HX and the closest nitrogen and boron atoms of fulborene). The overlap integrals over atomic basins were taken from the output of the AIMAll program. In the table only the largest absolute values of both kinds of indices are listed in Table 7. It turned out that in all cases the 3-center N-H-X index is the largest one, while the B-H-X index was always smaller by one order of magnitude and is therefore not included in the table. The δ (X, N, H) value is of order of 0.2·10−2 to 0.4·10−2 , what is the same order of magnitude as a typical index for two nitrogen and one boron atoms from the B12 N12 nanocage, which amounts to 0.58·10−2 and 0.85·10−2 for an N-B-N triad being a part of hexa- and tetragon, respectively. The examination of the largest 4-center index involving the halogen and the hydrogen atoms reveals more interesting features. First of all for some complexes the largest index is the δ (X, B, N, H) one, while for other complexes one of the δ (X, N, N′ , H) indices has the largest absolute value. The value with two nitrogen atoms is the largest for almost all iodine-containing complexes with the exception of the halogen-bonded case I, while for bromine- and chlorine-containing complexes



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this is the halogen-bonded type, for which the 4-center index with two nitrogen is the largest. The ratio between the largest and the smallest values of this index, equal to 15, is considerably higher than for the case of 3-center indices. In order to judge correctly the importance of the 4center index let us say that its typical value for atoms belonging to one tetragon of the B12 N12 nanocage amounts to 0.65·10−4 . By comparing these indices with the SAPT interaction energies from Table 2 one can notice that usually an increasing value of the index corresponds to a growing attraction between the fulborene and the hydrogen halides. Only when the interaction energies start to be very close to each other (the iodine case and the types III to V for bromine) the ordering of the largest 4-center indices becomes to be different from the energetic order of the complexes. Interestingly, these indices seem also to recognize the hydrogen bond, both proper (type II) and suboptimal (type IV) for the chlorine and bromine cases. For iodine the differences between the indices are much smaller, what can be explained by the larger Van der Waals radius of this atom, resulting in more uniform overlaps with the neighbors. For the hydrogen-bonded complexes a large value of δ (X, B, N, H) indicates a significant charge delocalization within these four atoms (it should be noted that in this case a similar value of the index occurs also for two other boron atoms joining the nitrogen atom closest to the HX molecule).

Natural Bond Orbital analysis The natural bond orbital (NBO) method 74,75 belongs to a group of models which allow to “translate" the quantum chemistry language to a form familiar for a general chemist, namely by identifying an NBO as an orbital occupied by a localized electron pair in the idealized Lewis structure. The NBO analysis begins from natural atomic orbitals (NAOs) produced from some 1-RD, which can be obtained from the HF, DFT or a more advanced theory. These orbitals are first combined within the same atom into natural hybrid orbitals (NHOs), which in turn are used to create NBOs as linear combinations of the NHOs belonging to adjacent atoms A and B,

σAB = CA (NHO)A + CB (NHO)B . 32 ACS Paragon Plus Environment

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If the absolute values of linear coefficients in Eq. (6) are not equal, it is said that the electron density is shifted towards one of two atoms, and the quantity %POL(A) = 100 | CA |2 , defined as a bond polarization of the A−B bond, is the quantitative measure of this effect. The bond polarization can be influenced by the interaction with another molecule or with environment. In the case of the intermolecular complex, these changes (the so-called repolarization of the bond) can provide us with an information about deformations of the electron cloud caused by the interacting partner, which in addition should be dependent on the relative positions of both molecules under study. An effective charge on the atom C (qC ) is another way of measuring the repolarization. In this section we examine this effect for the case of the NBO describing the σ bond in hydrogen halides. The corresponding data can be found in the first four data columns of Table 8, where the effective charges and polarization on halogen and hydrogen atoms in isolated HX molecules and in the complex are listed. In addition to the analysis of deformations, which take place within the NBOs of one monomer after the complexation, the NBO method provides several tools for measuring the charge transfer (CT) effect from one monomer to another, like the second-order energy describing a pairwise "interaction" between NBO orbitals. In order to obtain this quantity, one considers a donor NBO, χD , of the first molecule (a donor orbital has usually a large electron occupation number, denoted as nD ) and an acceptor NBO (χA∗ ) of the second molecule (an acceptor orbital has a small electron occupation number, i.e. it is an antibonding orbital). In the noninteracting limit we neglect the nondiagonal elements of the Fock matrix expressed by the NBOs, and the diagonal elements of this matrix can be treated as orbital "energies" (hχD |Fˆ χD i = EχD and hχA∗ |Fˆ χA∗ i = EχA∗ ). The interaction in this orbital pair is reintroduced by treating the corresponding nondiagonal element of the Fock matrix as a perturbation. By applying a standard perturbation theory to this two-dimensional problem one obtains the second-order NBO energy correction (called also the CT energy), which describes the energy lowering of the occupied level caused by the interaction ENBO (χD → χA∗ ) = −nD

|hχA∗ |Fˆ χD i|2 , EχA∗ − EχD


(7)


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while the CT itself can be estimated from the ratio of the nondiagonal Fock element and the difference of the diagonal elements. A large absolute value of the second-order energy from Eq. (7) indicates a significant CT effect. In the case of the B12 N12 :HX complexes both monomers incorporate typical Lewis acids (H in HX and B in B12 N12 ) as well as typical Lewis bases (X in HX and N in B12 N12 ) in the interaction site, so we can expect simultaneous interactions of NBOs in both directions and thus examine two pairs of potentially interacting NBOs. The first NBO pair consists of the ηN orbital describing a lone pair of the nitrogen atom closest to the HX molecule, and the lowest antibonding orbital of ∗ orbital. The second pair is composed of the η orbital, which hosts a lone pair of the HX, σHX X ∗ orbital, i.e. the antibonding σ orbital of B12 N12 . Absolute values of halogen atom, and the σN−B

the second-order energies (only those which are significantly different from zero) are presented in Table 8. It should be noted that for a delocalized network of bonds, the search for the optimum division into NBOs is not trivial and sometimes similar complexes end up with a different number of orbitals of a given type. The results shown in Table 8 were obtained with a modified occupancy threshold to assure that all fifteen complexes and the isolated B12 N12 have twelve ηN orbitals. The delocalized character of the bond network, which in the NBO picture is shown by significant (−40 kJ/mol) second-order energies for the σN−B → σN∗ ′ −B′ interaction (where N′ − B′ and N − B are two different pairs of atoms), allows to explain a seemingly strange finding (not shown in the table) that the occupation number of the lone pair orbital of N close to the HX molecule is larger than for the neighbour nitrogens. For an isolated nitrogen atom one would expect a decrease of the occupation, because the value of the CT energy indicates a direction of the charge transfer from N to HX. However, if the nitrogen atom is a part of an interconnecting network of atoms, it serves as a medium which transfers the electron charge from the whole fulborene. It is also interesting to note that the NBO analysis of the second-order energies supports the statement made during the disccussion of the energetics of the complexes about the importance of the 3d orbitals of bromine in the interaction energy. It turns out that the interaction between σN−B and a Rydberg orbital of Br composed in 35% from d orbitals has a sizable CT energy of 0.4 kJ/mol.


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Halogen-bonded structures From the geometry analysis we know that the type I configurations are halogen-bonded intermolecular complexes, i.e. the monomers are connected through the halogen-nitrogen bridge N· · · X with the attraction decreasing in order: I>Br>Cl (see Table 2). Within the NBO picture the change of strength of the N· · · X bonding is correlated with a different degree of the sp hybridization in ∗ antibondthe NHO halogen orbital, which is used for a construction of the σHX bonding and σHX

ing orbitals. Since the p orbital from the same shell overlaps more effectively with orbitals from another monomer than the s orbital, a higher contribution of p-type orbitals in the hybrid NHO should result in an increasing strength of the noncovalent interaction. The percentage of the s-type orbitals, denoted as %s(X), decreases indeed for heavier halogens, what agrees with the energetic ∗ orbital, which is responsible for intrends observed in Table 2. Therefore the antibonding σHX

termolecular interactions, is better "adapted" for closed-shell interactions for the case of heavier halogens: the emptier pz orbital is, the stronger is the intermolecular interaction because of the expected larger CT effect to the (almost) empty pz orbital. A comparison of the interaction energies from Table 2 and the ENBO values leads to the conclusion that there is a quasi-linear correlation between these two quantities, what would point out that the ENBO energies could be used to estimate the strength of interaction for a series of similar complexes. However, as the results for hydrogen-bonded complexes show, such dependence does not persist in all cases (vide infra). Interestingly enough, we do not see any repolarization effect (measured as %POL(X)) for Cl and Br in comparison to the isolated molecules. Since it has been stated in various studies 69,74,152–155 that the repolarization and the CT are two faces of the same process, we examined our results from this perspective. We see that indeed the highest |ENBO | (for B12 N12 :HI(I)) does correspond to the largest change in polarity in comparison to the isolated hydrogen iodide, but this trend is lost for both lighter halogens: the sizable secondorder energy measuring the CT effect is accompanied by virtually identical polarizations of HCl and HBr in the complex and in isolation. We can therefore conclude that the repolarization is not the only factor determining the CT in the intermolecular complexes. 35 ACS Paragon Plus Environment


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Hydrogen-bonded structures For the hydrogen-bonded complexes the main interaction, according to the NBO approach, occurs ∗ orbitals. The second-order energies describing the CT from nitrogen of between the ηN and σHX

B12 N12 to the antibonding orbital of HX are in this case several times larger than the energies for other structures, what agrees qualitatively with the magnitude of the interaction energies predicted by SAPT(DFT) and other theories. It should be noted, however, that the ratios of ENBO ’s for configurations of type I and II are much larger than the corresponding ratios for interaction energies (e.g. for complexes with HCl the NBO ratio is about 16, and for the interaction energies – about 4). Summarizing, the second-order NBO energies can be useful to explain some trends and main features of the interaction, but are not sufficient for a quantitative explanation of the observed trends in the intermolecular interactions. A word of caution should be said here about possible pitfalls in the effective charge analysis for the intermolecular interactions. Let us look at effective charges obtained from the NBO method for the case of the B12 N12 :HX(II) structures. Their examination shows that the hydrogen atom becomes more electropositive, and the halogen atom – more negative upon the complexation. On the other hand the large values of the second-order NBO energies signify a large electron CT from B12 N12 to HX. However, the total net charge (qX + qH ) of HX in the complex is positive for the case of HCl, while – according to the largest absolute value of ENBO in the whole table – the CT should occur in the opposite direction. On the other hand, the Bader effective charges for HCl sum, as expected, to a negative quantity in this case. The creation of the hydrogen bond between the hydrogen halide and the fulborene molecule can be also – somewhat artificially – represented as a superposition of two competing effects: the so-called hyperconjugation 74,75,156,157 and rehybridization. 158–160 The first effect is the transfer of ∗ orbital of a charge from a lone pair of a Lewis base (nitrogen in our case) to the antibonding σHX

Lewis acid (here: the HX molecule). Of course, the hyperconjugative effect alone tends to elongate the H–X bond because of the increased occupation of the antibonding orbital. On the other hand, the s-character of the hybrid orbital of halogen increases as a consequence of the CT from nitrogen 36 ACS Paragon Plus Environment

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of B12 N12 . This second effect, which is called rehybridization, can be explained by a different degree of overlap of the s and p orbitals of X with the 1s orbital of H, and it tends to shorten the H−X bond in comparison to the isolated hydrogen halide molecule. A favorable direction of the orbital change under complexation can be anticipated from the analysis of contributions of the corresponding s and p orbitals to the ENBO energy (e.g. for the chlorine case: 3sCl and 3pCl orbitals). It turns out that this energy will become more negative if the s character of the (NHO)Cl is amplified. (It should be noted parenthetically that this statement is in accordance with the Bent’s rule, 152 which says that atoms maximize the s character of the hybrid orbital aimed towards electropositive substituents.) The examination of Table 1 and Table 8 allows to make a conclusion that although the s-character of the H–X bonds in complexes is larger than in the isolated hydrogen halides, all H−X bonds have been elongated in the complexes, i.e. the hyperconjugative effect is more important than rehybridization in the case of HX complexes with fulborene. The structure types IV for Cl and Br represent the case of the interaction of the nitrogen atom with HX from the hydrogen side, but with the N−H−X angle significantly (about 30 degrees) different from 180. For the HI case the less electronegative iodine prefers to attach to the B12 N12 from the iodine side and we were not able to find a minimum analogous to the lighter halogen case. Therefore the reported nonzero CT energies are different for these two situations, i.e. there is an interaction of the lone pair of nitrogen with the antibonding HX orbital for Cl and Br, and the interaction of the lone pair of iodine with the B−N antibonding orbital. It should be noted that the second-order energy decreases in half when going from Cl to Br, what might explain the instability of the analogous configuration for iodine.

Four-center intermolecular interaction The configurations of B12 N12 :HX(III) and B12 N12 :HX(V) are of the same type, i.e. in both cases the HX molecule interacts with one boron and one nitrogen atom. In these configurations, the hydrogen and halogen atoms of HXs can both contribute to the stabilization of the complexes by



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acting as a Lewis acid and a Lewis base, respectively. This has its illustration in the table, where ∗ ∗ and from ηCl to σN−B and analogously for other halogens) both CT energies (from ηCl to σN−B ∗ ) is are simultaneously nonzero. The size of both effects is comparable, e.g. ENBO (ηCl → σBN ∗ ) is equal to −3.0 kJ/mol. Therefore, the composite equal to −5.2 kJ/mol, while ENBO (ηN → σHCl

mechanism of the interaction can be proposed for the configurations of the type III and V, where first the lone pair of nitrogen from the BN cage transfers some negative charge towards the HX molecule, and the excessive charge on HX is then redistributed towards the halogen atom, what as a result increases the back-CT towards the BN cage. Because of the resonance cooperation of two NBO pairs more charge exchange is possible than in the case of the interaction between one pair of NBOs, what leads to the stability increase of these two structures. This mechanism is much less pronounced for the B12 N12 :HI(V) case, where the CT energy from the lone pair of iodine to the ∗ ) energy. Similarly as for the antibonding B−N orbital is much smaller than the ENBO (ηN → σHI

hydrogen-bonded case, the hybrid orbital of the halogen in the complex as a larger admixture of the s-type orbital than in the isolated HX, what is in accordance of the Bent’s rule.

Summary and Conclusions We investigated the energetics and chemical nature of minimum structures for complexes of hydrogen halides with the B12 N12 nanocage. The geometry optimization performed with the M06-2X density functional and the aug-cc-pVDZ basis set revealed five local nonequivalent minima in each case. Among them the lowest minimum has a hydrogen-bonded character: the bond occurs between the hydrogen atom of the hydrogen halide and one of the nitrogen atoms of the B12 N12 molecule (II type). The minima of type IV have also the partial hydrogen-bond character for the HCl and HBr case, while the minima of types III and V represent a concerted interaction of the HX molecule placed almost parallel to one of the B−N bond with both N and B atoms. Finally the highest energetically minima of the type I have a halogen-bond character. The QTAIM analysis of the critical points for one-electron densities of these complexes confirms that the intermolecular


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BCPs correspond indeed to closed-shell interactions. Unfortunately, in the complexes of type of III and V simultaneous H · · · N and X · · · B bond paths never occur, contrary to our expectations. In place of expected BPs the QTAIM approach detects a link between the nitrogen and halogen atom. Two-, three-, and four-center delocalization indices confirm the concerted character of the interaction for these two structure types. Unexpected X · · · B bond paths detected by the Atomsin-Molecules program suggest that a usage of the term line path instead of bond path should be preferable in order to avoid the misinterpretation of the QTAIM results. 133 The reported stabilization energies, calculated by adding the ZPVE and the deformation energies to the SAPT interaction energy, remain negative for all but one case, indicating the stability of the structures under study. A sole exception is the case of the halogen bond for the lightest HX molecule. The energetic order of the stabilization energies remains the same as this of the interaction energies for the chlorine and bromine cases, while for the B12 N12 :HI complex the binding energy predicts a different most stable complex (the mixed type III). It should be noted, however, that for iodine the differences between various interaction energies are so small that the energetic order cannot be established unanimously. The SAPT analysis shows that the complexes of B12 N12 with HX are mostly stabilized by dispersion, with some netto contribution from induction for hydrogen-bonded complexes. The total first-order contribution is repulsive because of the large first-order exchange energy. The second-order induction energy increases in absolute value as the atomic number of halogen becomes higher. For all complexes the H−X distance is larger than in the isolated HX, what means that the vibrational spectrum of HX in the complex would be red-shifted. The NBO analysis shows a small rehybridization in the HX molecule upon the complexation which is compensated by the hyperconjugative effect. The SCS-MP2 interaction energies show an overall good agreement with the SAPT(DFT) results, while the MP2 interaction energies systematically overbind for all complexes under study. The latter fact has been already reported several times in the literature and it signifies that for the interaction dominated by dispersion it is neccessary to describe this component on a more advanced level than it is possible through the MP2 approach. The analysis of various orbital-freezing



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schemes for the B12 N12 :HBr complexes reveals that the 3dBr orbitals should be allowed to participate in the correlation energy in order to avoid errors of order of up to 4% in the interaction energy. Finally, it is important to emphasize that the necessity of a proper inclusion of the dispersion component sets high computational demands on investigations of intermolecular complexes involving the BN-nanostructures. Complexes of B12 N12 with hydrogen halides can be regarded as simple models allowing to predict a behaviour of more complicated BN nanostructures. The orientations of the hydrogen halide molecules towards B12 N12 show preferable sites for interactions on the surfaces of crystals built from BN units. They also allow to predict probable patterns of the B12 N12 assembly, which is of importance in crystal engineering. In addition, due to high directionality and tunable strength of halogen bonding, the capability of this simplest fulborene to participate in the halogen bonding is interesting in optics and electronics. 161–163

Acknowledgement This research was supported in part by PL-Grid Infrastructure. S.Y. acknowledges the scholarship from the Ministry of Science, Research and Technology of Iran for his stay at the University of Warsaw, Poland.

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Supporting Information Available The Supporting Information includes: (i) a figure depicting the dependence of the SAPT components on the intermolecular distance for the B12 N12 :HCl(III) case; (ii) detailed results for alternative optimization methods; (iii) a SAPT(DFT) curve for the B12 N12 :HCl halogen-bonded structure and for varying distance between chlorine and the nearest nitrogen atoms; (iv) detailed SAPT and supermolecular results for all utilized basis sets; (v) the QTAIM parameters for 1-RDs obtained with the HF and M06-2X methods; (vi) the figure of the bond paths for nonequilibrium geometries related to the B12 N12 :HI(III) structure. This material is available free of charge via the Internet at http://pubs.acs.org/.


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Table 1: Selected geometrical parameters for the M06-2X/aug-cc-pVDZ optimized structures of the B12 N12 :HX complexes and for isolated hydrogen halides. Distances are in Å and angles in degrees.

HCl B12 N12 :HCl(I) B12 N12 :HCl(II) B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) HBr B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V) HI B12 N12 :HI(I) B12 N12 :HI(II) B12 N12 :HI(III) B12 N12 :HI(IV) B12 N12 :HI(V)

RH−X 1.289 1.290 1.303 1.293 1.300 1.294 1.425 1.427 1.439 1.426 1.431 1.430 1.614 1.618 1.623 1.618 1.621 1.619

∆RH−X RX...N RH...N RX...B 0.001 0.014 0.004 0.011 0.005

3.189

0.002 0.014 0.001 0.006 0.005

3.216

0.004 0.009 0.004 0.003 0.005

3.281

ΘNHX

ΘNXH

ΘBXH

179.59 2.034 2.602 2.120 2.446

3.013 3.055

179.61 114.63 155.58 123.42

71.36 63.44 176.28

2.071 2.871 2.319 2.648

3.118 3.153

173.74 104.95 140.87 114.89

76.81 67.81 179.29

2.245 2.698 2.676

3.413 3.397 3.433

179.63 117.99 119.42

63.10 81.84 61.20

To facilitate the comparison of the interatomic distances with the sum of atomic radii we list Van der Waals radii of all utilized atoms from Table 1 of Ref.: 117 hydrogen (1.20), boron (1.65), nitrogen (1.55), chlorine (1.75), bromine (1.85), and iodine (1.98).



Table 2: Total CBS-extrapolated SAPT(DFT), MP2, and SCS-MP2 interaction energies for the minima of the B12 N12 :HX complexes. The stabilization energies obtained by adding the deformation and ZPVE to the SAPT interaction energy from the second column are also presented. The X/Y symbol is the abbreviation of aug-cc-pVXZ/aug-cc-pVYZ or of aug-cc-pCVXZ/augcc-pCVYZ and shows which basis sets were used for the CBS extrapolation. 98,99 Energies are in kJ/mol.

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Basis Energies Configuration B12 N12 :HCl(I) B12 N12 :HCl(II) B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V) B12 N12 :HI(I) B12 N12 :HI(II) B12 N12 :HI(III) B12 N12 :HI(IV) B12 N12 :HI(V)

ESAPT int D/T −5.58 −18.09 −11.89 −15.23 −12.14 −7.79 −15.50 −11.00 −10.83 −10.68 −12.38 −12.08 −11.52 −10.87 −11.24

aug-cc-pVXZ EMP2 ESCS−MP2 int int D/T T/Q D/T T/Q −6.94 −6.97 −4.74 −4.78 −20.25 −20.74 −15.89 −16.54 −15.58 −15.72 −10.95 −11.20 −17.67 −18.08 −13.33 −13.87 −15.77 −15.96 −11.10 −11.40 −9.77 −9.71 −7.00 −6.97 −19.21 −19.29 −14.33 −14.53 −15.77 −15.86 −10.52 −10.73 −14.58 −14.81 −9.53 −9.88 −15.30 −15.33 −10.36 −10.50 −13.77 −13.84 −10.26 −10.33 −14.96 −15.65 −10.51 −11.27 −15.47 −15.96 −9.95 −10.48 −14.09 −14.41 −9.49 −9.82 −15.02 −15.51 −9.69 −10.21

ESAPT int D/T −5.33 −17.24 −11.89 −14.51 −11.70 −7.64 −15.48 −11.00 −10.72 −10.72

aug-cc-pCVXZ EMP2 int D/T T/Q −7.04 −7.07 −20.39 −20.86 −15.80 −15.85 −17.81 −18.95 −15.99 −16.09 −9.92 −19.13 −16.13 −14.88 −15.65 −14.31 −15.47 −16.38 −14.84 −15.89

ACS Paragon Plus Environment

ESCS−MP2 int D/T T/Q −4.83 −4.84 −15.98 −16.64 −11.12 −11.31 −13.44 −13.95 −11.26 −11.51 −7.12 −14.12 −10.79 −9.76 −10.62 −10.67 −10.90 −10.67 −10.08 −10.37

ESAPT stab 0.80 −10.44 −6.29 −8.75 −6.39 −1.55 −8.05 −5.65 −4.01 −3.91 −5.25 −5.72 −6.93 −3.62 −4.87

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(2)

(2)

Table 3: Components of the SAPT(DFT) energies for the minima of the B12 N12 :HX complexes. The Edisp and Eexch−disp terms are extrapolated using the aug-cc-pVDZ/aug-cc-pVTZ CBS formula, 98,99 while all other values were calculated with the aug-ccpVTZ basis. Energies are in kJ/mol.

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Configurations/Energies B12 N12 :HCl(I) B12 N12 :HCl(II) B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V) B12 N12 :HI(I) B12 N12 :HI(II) B12 N12 :HI(III) B12 N12 :HI(IV) B12 N12 :HI(V)

(1)

Eelst −6.22 −25.76 −19.72 −20.51 −20.50 −9.89 −24.34 −19.25 −15.67 −18.07 −14.07 −15.84 −15.00 −12.15 −14.51

(1)

Eexch 11.30 38.67 33.81 33.49 34.71 16.14 40.22 35.95 29.94 33.96 18.56 26.81 29.82 24.01 28.59

(2)

Eind −6.61 −17.92 −18.63 −14.98 −18.43 −16.78 −19.60 −24.09 −14.49 −24.55 −26.77 −11.43 −20.74 −19.46 −19.84

(2)

Eexch−ind 6.24 11.25 15.03 9.91 14.54 15.97 14.07 20.58 11.83 21.19 19.84 7.80 16.39 15.56 15.60

(2)

Edisp −10.29 −18.84 −22.37 −18.78 −22.09 −13.21 −20.33 −24.33 −21.69 −23.52 −16.90 −17.03 −23.94 −20.85 −22.94


(2)

Eexch−disp 0.97 2.49 3.02 2.33 2.96 1.36 2.72 3.34 2.58 3.27 1.77 2.00 3.07 2.69 2.90

δ EHF −0.97 −7.98 −3.03 −5.69 −3.33 −1.38 −8.24 −3.20 −3.33 −2.96 5.19 −4.39 −1.12 −0.67 −1.04

ESAPT int −5.58 −18.09 −11.89 −15.23 −12.14 −7.79 −15.50 −11.00 −10.83 −10.68 −12.38 −12.08 −11.52 −10.87 −11.24


(2)

(2)

Table 4: Components of the SAPT(DFT) energies for the minima of the B12 N12 :HX complexes. The Edisp and Eexch−disp terms are extrapolated using the aug-cc-pCVDZ/aug-cc-pCVTZ CBS formula, 98,99 while other values were calculated with the aug-ccpCVTZ basis. In parenthesis the analogously extrapolated SAPT(DFT) results obtained in the aug-cc-pVTZ basis with frozencore (1s2s2p for Cl and 1s for B and N) are shown. All energies are in kJ/mol. Configurations/Energies B12 N12 :HCl(I) B12 N12 :HCl(II)

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B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V)

(1)

Eelst −6.57 (−6.54) −25.85 (−25.91) −20.03 (−20.06) −21.61 (−21.66) −20.80 (−20.83) −9.79 −24.30 −19.21 −15.55 −18.05

(1)

Eexch 11.99 (12.04) 39.48 (39.52) 34.60 (34.66) 34.23 (34.27) 35.52 (35.57) 16.13 40.24 35.93 29.94 33.93

(2)

Eind −8.05 (−8.10) −18.95 (−19.18) −19.81 (−20.03) −15.92 (−16.10) −19.69 (−19.90) −16.85 −19.61 −24.07 −14.47 −24.54

(2)

Eexch−ind 7.69 (7.76) 12.42 (12.63) 16.19 (16.41) 10.96 (11.12) 15.82 (16.02) 16.04 14.09 20.58 11.83 21.19

(2)

Edisp −10.43 (−10.39) −18.98 (−18.95) −22.53 (−22.44) −18.91 (−18.15) −22.25 (−22.16) −13.30 −20.40 −24.38 −21.73 −23.57


(2)

Eexch−disp 1.03 (1.00) 2.58 (2.53) 3.09 (3.01) 2.40 (2.25) 3.03 (2.96) 1.39 2.74 3.35 2.59 3.28

δ EHF −0.99 (−0.97) −7.94 (−7.98) −3.40 (−3.03) −5.66 (−5.69) −3.33 (−3.34) −1.26 −8.24 −3.20 −3.33 −2.96

ESAPT int −5.33 (−5.20) −17.24 (−17.34) −11.89 (−11.48) −14.51 (−13.97) −11.70 (−11.68) −7.64 −15.48 −11.00 −10.72 −10.72

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Table 5: MP2 interaction energies obtained with aug-cc-pVTZ and aug-cc-pCVTZ basis sets and with a) all electrons correlated, MP2 MP2 EMP2 int (AE), b) frozen-core including 3d orbitals of bromine, Eint (C1), c) frozen-core excluding 3d orbitals of bromine, Eint (C2). ∆EMP2 int (Cn) denotes differences between the all-electron and frozen-core results. All energies are in kJ/mol.

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Configurations/Energies ESCF int Basis B12 N12 :HBr(I) 2.50 B12 N12 :HBr(II) −1.74 B12 N12 :HBr(III) 6.02 B12 N12 :HBr(IV) 5.86 B12 N12 :HBr(V) 5.50 Basis B12 N12 :HBr(I) 2.46 B12 N12 :HBr(II) −1.61 B12 N12 :HBr(III) 6.03 B12 N12 :HBr(IV) 5.88 B12 N12 :HBr(V) 5.51

MP2 MP2 EMP2 int (AE) Eint (C1) ∆Eint (C1) aug-cc-pVTZ −9.13 −8.96 −0.17 −17.86 −17.25 −0.61 −14.59 −14.10 −0.49 −13.57 −13.12 −0.45 −14.19 −13.75 −0.44 aug-cc-pCVTZ −9.26 −9.01 −0.25 −18.11 −17.24 −0.87 −14.91 −14.15 −0.76 −13.84 −13.14 −0.70 −14.49 −13.78 −0.71


MP2 EMP2 int (C2) ∆Eint (C2)

−9.03 −17.88 −14.50 −13.53 −14.09

−0.10 0.02 −0.09 −0.04 −0.10

−9.16 −18.15 −14.87 −13.85 −14.42

−0.10 0.04 −0.04 0.01 −0.07


Table 6: QTAIM parameters for the intermolecular BCPs from the MP2 charge density. The aug-cc-pVDZ basis has been used. The results are given in a.u.

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Configurations B12 N12 :HCl(I) B12 N12 :HCl(II) B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V) B12 N12 :HI(I) B12 N12 :HI(II) B12 N12 :HI(III) B12 N12 :HI(III) B12 N12 :HI(IV) B12 N12 :HI(V)

BCP (N|Cl) (N|H) (N|Cl) (N|H) (N|H) (N|Br) (N|H) (B|Br) (N|H) (N|Br) (N|I) (N|H) (N|H) (N|I) (B|I) (N|H)

ρBCP 0.0081 0.0235 0.0113 0.0202 0.0126 0.0092 0.0225 0.0111 0.0147 0.0104 0.0104 0.0166 0.0089 0.0089 0.0088 0.0091

∇2ρBCP 0.0274 0.0632 0.0315 0.0532 0.0387 0.0290 0.0578 0.0265 0.0376 0.0266 0.0301 0.0388 0.0265 0.0209 0.0197 0.0268

λ1 −0.0049 −0.0262 −0.0075 −0.0210 −0.0112 −0.0052 −0.0241 −0.0062 −0.0134 −0.0063 −0.0057 −0.0156 −0.0067 −0.0048 −0.0038 −0.0068

λ2 −0.0049 −0.0261 −0.0028 −0.0202 −0.0055 −0.0052 −0.0239 −0.0035 −0.0121 −0.0020 −0.0057 −0.0155 −0.0024 −0.0014 −0.0021 −0.0031

λ3 0.0372 0.1155 0.0418 0.0943 0.0554 0.0394 0.1058 0.0362 0.0631 0.0349 0.0415 0.0699 0.0357 0.0272 0.0256 0.0367


GBCP 0.0059 0.0149 0.0073 0.0129 0.0088 0.0063 0.0138 0.0062 0.0093 0.0062 0.0067 0.0096 0.0058 0.0048 0.0045 0.0058

VBCP −0.0049 −0.0141 −0.0067 −0.0124 −0.0080 −0.0053 −0.0131 −0.0061 −0.0092 −0.0057 −0.0059 −0.0095 −0.0048 −0.0043 −0.0040 −0.0050

HBCP −GBCP /VBCP 0.0010 1.2041 0.0080 1.0567 0.0006 1.0890 0.0005 1.0403 0.0008 1.1000 0.0010 1.1887 0.0007 1.0534 0.0003 1.0492 0.0001 1.0109 0.0005 1.0877 0.0008 1.1356 0.0001 1.0105 0.0010 1.2083 0.0005 1.1163 0.0005 1.1250 0.0008 1.1600

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Table 7: The electron delocalization indices computed at the HF and MP2 levels. The the aug-cc-pVDZ basis has been used.

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Configurations/Method B12 N12 :HCl(I) B12 N12 :HCl(II) B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V) B12 N12 :HI(I) B12 N12 :HI(II) B12 N12 :HI(III) B12 N12 :HI(IV) B12 N12 :HI(V)

δ (X, N) HF MP2 0.074 0.077

δ (X, B) HF MP2

0.062

0.060

0.015

0.025

0.063 0.090

0.060 0.094

0.013

0.022

0.059

0.058

0.017

0.028

0.058 0.113

0.057 0.118

0.015

0.026

0.050 0.053 0.050

0.049 0.047 0.049

0.013 0.015 0.013

0.024 0.026 0.023

δ (N, H) HF MP2 0.091 0.029 0.074 0.040

0.100 0.032 0.085 0.045

0.098 0.021 0.064 0.034

0.107 0.023 0.070 0.037

0.085 0.040

0.092 0.043

0.041

0.045


|δ (X, N, H)| |δ (X, B, N, H)| |δ (X, N, N ′ , H)| HF HF HF 0.14·10−2 0.21·10−4 0.15·10−2 0.20·10−3 −2 0.26·10 0.47·10−4 −2 0.22·10 0.16·10−3 −2 0.29·10 0.54·10−4 −2 0.17·10 0.42·10−4 −2 −3 0.26·10 0.21·10 0.26·10−2 0.55·10−4 0.37·10−2 0.13·10−3 −2 0.34·10 0.73·10−4 −2 0.18·10 0.40·10−4 −2 0.42·10 0.60·10−4 −2 0.41·10 0.33·10−4 −2 0.14·10 0.14·10−4 −2 0.42·10 0.86·10−4


Table 8: The NBO analysis of the hydrogen halides in the complexes and in an isolated form of the M06-2X charge density calculated in the aug-cc-pVDZ basis. The NBO charges at hydrogen and halogen atoms (qH and qX , in a.u.), the polarization of the H–X bond, the s-character of the hybrid orbital of X involved in the H–X bond, and the second-order NBO energies for two possible interorbital interactions are presented. Energies are in kJ/mol.

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Molecules HCl B12 N12 :HCl(I) B12 N12 :HCl(II) B12 N12 :HCl(III) B12 N12 :HCl(IV) B12 N12 :HCl(V) HBr B12 N12 :HBr(I) B12 N12 :HBr(II) B12 N12 :HBr(III) B12 N12 :HBr(IV) B12 N12 :HBr(V) HI B12 N12 :HI(I) B12 N12 :HI(II) B12 N12 :HI(III) B12 N12 :HI(IV) B12 N12 :HI(V)

qX −0.283 −0.277 −0.314 −0.248 −0.304 −0.257 −0.210 −0.199 −0.238 −0.172 −0.210 −0.170 −0.097 −0.087 −0.119 −0.063 −0.048 −0.067

∗ ) −E ∗ qH %POL(X) %s(X) −ENBO (ηN → σHX NBO (ηX → σBN ) 0.283 64.39 15.59 0.282 64.31 15.19 1.80 0.294 65.88 18.35 37.41 0.304 65.44 16.22 2.97 5.18 0.295 65.62 17.79 25.08 0.305 65.57 16.58 6.15 1.96 0.210 60.37 13.43 0.204 60.49 12.92 3.21 0.219 62.07 15.62 37.16 0.229 61.72 13.93 3.93 4.35 0.217 61.44 14.55 11.62 0.230 61.72 13.86 3.18 2.59 0.097 55.10 10.66 0.086 54.73 9.99 5.73 0.109 56.20 11.97 22.36 0.116 56.06 11.00 3.51 2.55 0.108 55.57 10.52 2.17 0.117 56.11 11.02 3.76 0.92


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Figure 1: The studied orientations of the HX molecule on the surface of the B12 N12 nanocage. The scheme IVa shows the orientations of HCl and HBr with respect to the fulborene, while IVb presents the special case of HI.



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Figure 2: The M06-2X/aug-cc-pVDZ electrostatic potential on the 0.001 a.u. density isosurface for the isolated B12 N12 (top, labelled a), and HCl, HBr, and HI (the second row from left to right, b, c, and d, respectively). The position of the most positive values of MEP close to BCP are indicated by small black dots. For the case of B12 N12 they show the boron side of the nanocage and its π -hole, while for the HX molecules – the position of the σ -hole. The colour ranges denote: red – greater than 96, yellow – from 96 to 33, green – from 33 to 0, and blue – less than 0 kJ/mol.


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Figure 3: The contour map of the charge density of the B12 N12 :HCl(III) complex. Numbers on the isodensity lines are in a.u.



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Figure 4: The relief map of the Laplacian of the charge density of the B12 N12 :HCl(III) complex. The minimum and maximum ∇2ρ values on the relief map are −1.55 × 106 and 6.90 × 105 a.u., respectively. Red and yellow colours refer to the most and least negative values of ∇2ρ , while blue, cyan, and green colours denote the most to least the positive values of ∇2ρ .


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For Table of Contents use only


Structure and Energetics of Complexes of B12N12 with Hydrogen

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