Carboranes as Lewis acids: Tetrel bonding in CB11H11 carbonium ylide

High-level quantum-chemical computations (G4MP2) are carried out in the study of complexes featuring tetrel bonding between the carbon atom in the carbenoid CB11H11 obtained by hydride removal in the C-H bond of the known closo-monocarbadodecaborate anion CB11H12() and acting as Lewis acid (LA) and Lewis bases (LB) of different type; the electron donor groups in the tetrel bond feature carbon, nitrogen, oxygen, fluorine, silicon, phosphorus, sulphur and chlorine atomic centers in neutral molecules as well as anions H(), OH() and F(). The empty radial 2pr vacant orbital on the carbon center in CB11H11 , which corresponds to the LUMO, acts as a Lewis acid or electron attractor, as shown by the molecular electrostatic potential (MEP) and electron localization funcion (ELF). The thermochemistry and topological analysis of the complexes {CB11H11:LB} are comprehensively analyzed, and classified according to sharing or closed-shell interactions. ELF analysis shows that the tetrel C···X bond ranges from very polarised bonds, as in H11B11C:F() to very weak interactions as in H11B11C···FH and H11B11C···O=C=O.


Introduction
The very stable B12H12 (2-) dianion and its neutral dicarbon counterparts ortho-(1,2-C2B10H12), meta-(1,7-C2B10H12) and para-carborane (1,12-C2B10H12) are icosahedral systems that are closely related to elemental boron. Their isoelectronic analogue, closo-monocarbadodecaborate anion CB11H12 (-) , first prepared in 1967 [1] and further with other synthesis methods [2,3], is similarly resistant to cage degradation, and many derivatives have been synthesized as described in the literature [4]. The stability and three-dimensional aromaticity of CB11H12 (-) has also been explained using quantum-chemical computations [5]. Extraction of hydride H (-) in the C-H bond from CB11H12 (-) leads to a carbocation ylide or carbenoid (1) with a vacant radial 2pr orbital on the cage carbon atom as shown in Figure  1. On the other hand, reaction mechanisms of polyhedral (car)boranes and their derivatives are scarce in the literature and further research is needed in this respect [6][7][8]. Thus, the permethylated carbenoid analogue CB11Me11 has been postulated as a reaction intermediate during the extraction of the L substituent from L-CB11Me11 carboranes (L = BrCH2CH2 or (CF3)2CHO) by electrophiles [6,7], further reacting with arenes in the presence of (CF3)2CHOH to generate 1-aryl-CB11Me11 products [6][7][8]. The methyl groups in the permethylated anion CB11Me12 (-) have substantial CH3 (-) (methide) character according to DFT computations [6][7][8], and can easily bind to transition metal and main group elements. On the other hand, in recent years, tetrel bonding -defined as an interaction between any electron donating system (ED) and a group 14 element acting as Lewis acidhas called the attention of both experimentalists [9] and theoreticians [10,11]. Here the carbon center in (1) is clearly an acceptor of electrons or Lewis acid, hence we can define a tetrel bonding interaction with an electron donor (ED) or Lewis base, as shown in Figure  1c.
The goal of this work is to study the electronic interaction between the naked carbon vertex in the carbenoid (1) with a series of electron donor molecules and anions, leading to tetrel C-X bonds. The chosen 18 LB systems, including the anions H (-) , F (-) and OH (-) , are displayed below in Scheme 1 with the corresponding label.

Computational Methods
Electronic structure quantum-chemical computations were carried out using the G4MP2 model [12,13], which is a fourth-generation method available in the Gaussian16 (c) scientific software [14]. This method combines density-functional theory [15,16] and second-order perturbation theory [17] and provides an accurate and economical method for thermochemical predictions. The G4(MP2) model works as follows: The geometries of the molecules are optimized at the B3LYP/6-31G(2df,p) level of theory and then a series of single point energy calculations at higher levels of theory are computed. The zero-point energy, E(ZPE), is based on B3LYP/6-31G(2df,p) frequencies scaled by 0.9854, the same as in G4 theory. The first energy calculation is at the triples-augmented coupled cluster level of theory, CCSD(T), with the 6-31G(d) basis set, i.e., CCSD(T)/6-31G(d). This energy is then modified by a series of energy corrections to obtain a total energy E0. For more details on the G4(MP2) method the reader es referred to References [12,13]. In the particular case of the 1:LB complexes, we computed the enthalpy and free energy differences between the complex and separated systems 1 and LB at room temperature as indication of stability of the complex. All complexes included in this work correspond to energy minimum structures, checked with frequency computations. The quantum theory of atoms-in-molecules (QTAIM) [18,19] was used in the topological analysis of the electron density of the 1:LB complexes, with the scientific software AIMAll [20]. This method is based on the analysis of the electron density , its gradient  and the corresponding Laplacian  2 .
For further aspects of this methodology, the reader is referred to the above References [18,19] and Section 3.3.1 below. The electron localisation function (ELF) [21,22] was also used in the topological analysis of the complexes. The ELF is a distribution function which measures the probability of finding an electron near to another electron with an opposite spin, as further described in Section 3.3.2 below.

Geometries of complexes (1:n), n = 2-19
In Figures 2a-b we display the G4MP2 optimized geometries of CB11H12 (-) with C5v symmetry -the (1:2) complex -and the carbonium ylide CB11H11 (1), with the different tetrel complexes (1:n), n = 2-19, and structural parameters when necessary, in order to highlight the atomic rearrangements undergone due to the complexation process. The loss of H (-) in the C-H leads also to a structure with C5v symmetry -with a geometrical change which involves a considerable flattenning of the CB5 pentagonal pyramid with expansion of the corresponding B5 pentagon, since there is an increase of the B-B bond distance of  = +0.022 Å. As we proceed down the cage from the top (C atom), the B-B bond differences are +0.022 Å, +0.022 Å and -0.015 Å. Quite noticeable is the shortening of the apical intracage C···B distance (  -0.3 Å). The B-H bond distances hardly change at all, with a very slight shortening upon loss of H (-) with  = -0.007 Å, -0.008 Å and -0.001 Å, from top to bottom respectively. If we take the C and B cage nuclei as point charges and define a distorted C5v icosahedron, the corresponding volumes are V(CB11H12 (-) ) = 12.03 Å 3 and V(CB11H11) = 11.90 Å 3 and therefore there is a shrinkage of the cage by V = -0.13 Å 3 (1%). In summary the extraction of hydride in the C-H bond from CB11H12 (-) implies a minor change in the cage volume with a flattening of the top CB5 pentagonal pyramid and minor changes as we proceed down the cage from the top C atom. In Figures 2c-s we display the optimised geometries for the remaining complexes, with the coordinates gathered in the Sup. Info. (SI, Tables S1-S10).The shortest and longest C···X tetrel interactions correspond to the original anion CB11H12 (-) or complex (1:2) and the CB11H11···O=C=O tetrel complex (1:13), respectively. In the latter case the attractive interaction is clearly non-covalent in origin with d(C···O) = 2.693 Å, as will be discussed later on. We should also emphasize the C···X interaction of the CH2 and SiH2 complexes (1:3) and (1:16) respectively, with the LB groups tilted from the C5 axis of rotation. In all other systems, including the CF2 complex (1:4), the C···X bond is aligned with the C5 axis of rotation.   In Figure 3 we display the d(C···X) distances in the tetrel bonding complexes, ordered from shortest to longest. Clearly, we can classify 5 groups according to the C···X distances, in increasing order: (i) The original anion CB11H12 (-) or (1:2) complex, (ii) complexes with d  1.4-1.5 Å including complexes (1:k2), with k2 = (3-12, 14), (iii) complexes with d  1.8 Å, including complexes (1:k3), with k3 = 16-18, (iv) complexes with d  2.0 Å, including complexes (1:15) and (1:19), and finally (v) the (1:13) complex with d  2.7 Å.

Thermochemistry of complexes (1:n), n = 2-19
In this subsection we predict the ΔH and ΔG of the tetrel bonding complexes (1:n), n = 2-19. In Table 1 we gather the computed enthalpies and free energies at the G4MP2 level of theory. The free energy of formation for complexes (1:n) is always negative, except for (1:13) and (1:15) complexes, namely the O=C=O and FH complexes respectively. Small negative values (|G| < 10 kJ·mol -1 ) are obtained for complexes (1:6) and (1:19) with Lewis bases N2 and ClH respectively. Therefore, all complexes with negative G should be formed at room temperature spontaneoulsy, provided an isolated Lewis acid (1) approaches an isolated Lewis base. On the other hand, the enthalpies of formation are negative for all complexes, an indication that the bond energies of a given complex (1:n) have a lower value than the bond energies of separated systems (1) and (n). In order to better visualize the similarities and differences of the thermochemical aspects of complexes (1:n) we plot G and H vs. n in increasing order of each state function. As shown in Figure 4 from left to right in the abscissa, the G (black) and H (blue) in complex formation follow the same order as function of Lewis base number, except for ClH (1:19) and N2 (1:6) where the order is inverted in the H tendency as compared to G; this is due to the noncovalent and covalent character of each interaction. Hence, the formation of anionic complexes are the most energetic and favourable ones in the order  (1:7) follows with Lewis bases NH=CH2, PH3 and NH3 respectively. A smaller (positive) slope of G/H vs. n appears with complexes (1:5), (1:18), (1:9), (1:12) and (1:11) always in increasing order, which correspond to Lewis bases CO, SH2, NCH, O=CH2 , and OH2 respectively. The weakest bound complexes, with G < 0, correspond to (1:19) and (1:6) with Lewis bases ClH and N2 respectively. Finally complexes (1:13) and (1:15) with Lewis bases O=C=O and FH respectively show a predicted quantum-chemical value of G > 0 and therefore one should not expect a spontaneous formation of these complexes at room temperature. It is noteworthy to mention the tiny value H(1:15) = -0.7 kJ·mol -1 for FH attachment to (1); this number is within the accuracy of the method and therefore a heat of formation for complex (1:15), or the bond energy on both sides of the equation remains unaltered. (3) (3) In Figure 5 we show for (1), from left to right, the lowest unoccupied molecular orbital (LUMO), the molecular electrostatic potential (MEP) and the electron localization function (ELF). These electronic structure features are computed using the optimized geometry of the system with the G4MP2 method -B3LYP/6-31G(2df,p) model chemistry for structure optimization. As noticed in Figure 5a, the LUMO of the carbonium ylide has considerable wave function amplitude around the C atom -as compared to other regions of the molecule, an indication of the electron acceptor (Lewis acid) nature of the C ylide center; this property is confirmed by the shape of the MEP and the corresponding π-hole just on the top of C ylide center, where negative (surplus) electron density is attracted to.
The ELF from Figure 5c shows disynaptic V(B,H) yellow basins corresponding to the the B-H bonds, and the ELF distribution around the CB11 icosahedral cage can be partitioned into green disynaptic and trisynaptic basins as we will describe below in Section 3.3.2. The molecular electrostatic potential (MEP) is the potential energy of a proton at a particular location near a molecule. Negative electrostatic potential corresponds to an attraction of the proton by the concentrated electron density in the molecules. The MEP of (1) - Figure 5b -shows that the potential energy of a proton is most positive above the C atom, with a π-hole of +0.061 au, hence a repulsive region for a proton approaching (1), or electron acceptor region. The MEP is smoothly changing from positive to negative values of the potential energy as the proton moves from the C atom down to the B skeleton cage region. A proton would then be attached more favourably to the lower region of the carbonium ylide (1). In other words, Lewis bases, electron donors and nucleophiles should then tend to bind through the C atom of the ylide, hence the study of the tetrel bonding in the complexes (1:n). The Quantum Theory of Atoms in Molecules (QTAIM) [18][19] is a useful tool for analyzing the electronic structure of a polyatomic many-electron system, with the electron density (r) as the central function. The topological properties of (r) are analyzed with tha gradient ∇ ⃗ and Laplacian ∇ , with the eigenvalues of the latter being (1, 2, 3). The critical points are those with ∇ ⃗ = 0 ⃗ and a bond critical point (BCP) has 3 > 0 associated with the bond path direction, and 1 < 0, 2 < 0 the two latter associated to two directions where ∇ is a maximum; the BCP (-,-,+) appears at the intersection of the bond path with the interatomic surface S. Other critical points are classified according to the signs of i : Nuclei positions with (-, -, -); ring critical points with (-,+,+); cage critical points with (+,+,+). We should also introduce the local electron kinetic (G > 0), potential (V < 0 ) and total (H) energy densities, with H = G + V, also useful parameters at the BCP for the description of the type of bonding interaction between atoms in a many-electron system [23]. In the SI (Table S11)     In Figure 6a we plot the electron density at the BCP for the C····X interaction vs. d(CX) distance. The largest values of BCP correspond to CH2 ( BCP(CH2) = 0.32 e/a0 3 ), CF2 , and CO, followed by H (-) , OH (-) , NHCH2, N2 and F (-) . Another group follows with lower values, NH3, OCH2, PH3, OH2, SH2 and further down SiH2 and ClH with similar values. Finally, the lowest BCP correspond to FH and CO2, the latter with BCP(CO2) = 0.01 e/a0 3 . We should emphasize that the ratio BCP,max(CH2)/BCP,min(CO2) = 32 gives an idea of the topological differences in these BCPs. Given the different type of C···X interactions in the complexes, the BCP vs. d(CX) can be fit to an approximate negative exponential curve with BCP (dCX) = a + b·exp(c·dCX), with a = -0.022, b = +3.222 and c = -1.751, and a correlation factor of R 2 = 0.99 for closed-shell interacting complexes: CO2, FH, SiH2, N2 and NCH. This curve is displayed in the SI file ( Figure S1). In general, very good correlations appear if we fix the two interacting atoms, both belonging to the same row of the Periodic Table [24][25][26].
In order to estimate the type of interaction we need to go beyond the electron density at the BCP and analyze the second derivative, the Laplacian  2 , and the kinetic, potential and total energy, G, V and H respectively, of the BCPs in complexes (1:n). In Figure 6b the Laplacian is plotted vs. LB(n) in increasing order. Clearly, we can distinguish the shared interactions for  2  > 0 in the lower left corner and closed-shell interactions for  2  > 0 in the upper right corner of Figure 6b. The two-electron sharing in complexes with H (-) , CO, CF2 and CH2 is large, and it is diminished up to OH2. For the complex with HCl,  2  = -0.0067 e/Å 5 , namely in the limit between sharing and closed-shell interactions. For positive Laplacians, in increasing order, the LB in the (1:n) complexes correspond to: NCH, CO2, FH, SiH2 and N2; in these systems the closed-shell interactions are important.
A further analysis of the BCP in the C···X interactions of the (1:n) complexes can be found in the values of G, V and H -with H = G + V being the total energy -as displayed in Figure 6c. The kinetic energy G is associated with repulsion in the bonding region, and the potential energy V shows the effect of the electric field in the internuclear region. Thus, according to Figure 6c, the G and V profiles are inverted for N2, NCH, F (-) and OH (-) , but due to the nature of different nuclei in the C-X interactions, this is not always the case, as seen when we follow the profiles as function of LB(n). Thus, H can be used for characterising the covalent character of an attractive interaction betweens atoms [21], and a combination of the Laplacian and H for the characterisation of hydrogen bonds [22].

Electron Localisation Function (ELF) analysis of complexes (1:n), n = 2-19
In order to better understand, from the electronic structure point of view, the tetrel bonding in the (1:n) complexes, we further computed the Electron Localization Function (ELF) [27][28], a measure of the likelihood of finding an electron in the neighborhood space of a reference electron located at a given point and with the same spin; therefore ELF is a measure of the Pauli repulsion or exchange interaction [29][30]. The ELF for the carbonium ylidene CB11H11 (1) is shown in Figure 4c. ELF values ranges from 0 to 1 (normalized and without units). In the SI file we provide the ELF for all complexes (1:n) not shown here (Table S12) and below we have selected four cases with short, medium, long and very long C···X distances, according to  Table 2 we gather the ELF function for these four complexes. Below each ELF function of a given complex (1:n), we also report the function value and average population for the different types of basins. A threshold of 0.2 electrons is considered as to include or not a basin in a group. For example if a V(B,B) basin has a population of 0.8 electrons, the latter is not included in the group of the V(B,B) basins with a population around 1.5. In bold letters we report the value of the disynaptic basin corresponding to the tetrel C···X interaction.
In Table 2 we use the following notation: 74. We should emphasize that ELF is a function which reports the probability of finding an electron pair with opposite spins in a region of space. Using a certain isovalue, we are able to define regions of space, basins, with a certain probability to find an electron pair. For example, in the plots of the ELF, we used an isovalue of 0.83, in other words, we plot regions of space where we have a high probability to find a pair of electrons. Once the basins are defined, we can integrate the electronic density into those basins, which are the values reported above in Table 2, and correspond to the number of electrons in that basin. In ELF analysis the partition of space is not based on the electron density, as in AIM, but on the ELF probability function.
As reported in Table 2, the C···X interaction in the selected complexes can be described by the presence or absence of V(C,X) valence basins, and its population. For the complexes (1:6) and (1:17), these values are V(1:6)(C,N) = 2.62 and V(1:17)(C,P) = 2.21 respectively. Therefore, ELF describes the C···X for the N2 complex as a bond, with a multiplicity close to 1.5, between the C(ylide) and N nuclei, and for the PH3 complex a C(ylide)P single bond, with additional 0.2 electrons. As regard to the (1:13) complex with CO2, the ELF does not localize a basin between the C(ylide) and the O=C=O molecule, and thus no bonding is expected neither a complex formation. The lone pairs from N2 and CO2 appear as red monosynaptic basins, as displayed in Table 2.

Discussion
The presence of a filled or empty lone pair on the C atom in the known anion CB11H12 () , complex (1:2), depens on whether we remove a proton or a hydride from the C-H bond, leading to a dianion [H11B11C:] (2) (1b) or a carbonium ylide H11B11C (1), respectively, the latter process is shown in Figure 1. The tetrel complexes (1:n) presented in the previous section show a rich variety of thermochemical and electronic structure features, with tetrel C···X interactions from different nuclei: X = {H, C, N, O, F, Si, P, Cl}. The C(ylide) center in (1) confers to this particular molecule with a Lewis acid (LA) character, hence the tetrel denomination. The strength for electronic attachment in (1) is given by the computed free energy of formation (1) + (n)  (1:n). The strongest complexes correspond to those formed with anions H () , OH () and F () , and the weakest complexes to those formed with FH, CO2, N2 and ClH. The C···X distance varies considerably in all complexes, ranging from 1.081 Å for (1:2), LB = H () , to 2.694 Å for (1:13), LB = CO2. The complex strength is not related to the C···X distance, namely, complexes with similar C···X distances may have different free energy of formation; e.g. the free energy of formation for complexes (1:14) LB = F () and (1:6) LB = N2 is -530 kJ·mol -1 and -3 kJ·mol -1 respectively, with very similar d(C-X) distances, 1.364 Å and 1.375 Å.
Examples of recent related systems is the 3D analogue of phenyllithium, the lithiacarborane CB11H11:Li () , studied in solution, as a solid and by quantum-chemical computations [31]. Indeed, Li () is a very poor Lewis base, but certainly attaches to (1), as recently shown, and defined as the lithiated dianion [CB11H11] (2-) . On the other hand, this process can also be seen as a carborane dianion (1b) -a very reactive species -attached to Li (+) , a described in Reference [31].  (-) and therefore this is a rich field not only from a synthetic point of view but also for studying the electronic structure of tetrel C···X bonds in these compounds, and specially if the isolated tetrel complexes (1:n) could ever be synthesized, taking into acount that this work is purely theoretical with predictive quantum-chemical computations.
The electronic structure of the complexes has been analyzed thoroughly with AIM and ELF methods, showing the C···X sharing and closed-shell interactions in the complexes according to the values of the Laplacian of the electron density. In Table 3, we gather the ELF values for disynaptic basins V(C, X) in the C···X region showing values of ELF: we can find very polarised C-F bonds in (1:14) -only one electron in the C···X region -single C-X bonds for H (-) and NH=CH2, and intermediate cases, such as in complex (1:5) with a 1.5 multiplicity C-C bond for the CO complex. No V(C,X) disynaptic basins are found for CO2 and FH, an indication of the poor electron-donating ability of these Lewis bases (LB), with indeed long d(C-X) distances and positive free energies of formation G > 0, hence confirming the unlikely formation of these two complexes. shown in the SI file as Table S13. The shortest C(ylide)···N distance, d(C···N) = 1.477 Å, corresponds to the pyridine complex 1-(4-methoxypyridinium)-1-carba-closo-dodecaborane [33]. The longest C(ylide)···N distance, d(C···N) = 1.554 Å, corresponds to the complex 12-iodo-1-(4-pentylquinuclidine)-1-carba-closo-dodecaborane [34]. There is a tetrel complex of (1) with NH3, 1-amino-2-fluorocarba-closo-dodecaborane [35], where one B-H vertex hydrogen atom on position 2 has been substituted by a fluor atom, with d(C···N) = 1.486 Å. Our (1:7) tetrel complex H11B11C:NH3 has a predicted d(C···N) = 1.498 Å with the G4MP2 model, hence there is a good agreement with experiment.

Conclusions
The results presented in this work show that by means of quantum-chemical computations we can predict the formation of tetrel complexes between the icosahedral carbonium ylide CB11H11 -derived from extraction of H () in the known anion CB11H12 () -and a set of simple molecules and anions. The driving force of formation for these complexes can be accounted for from thermochemical quantum-chemical computations, using statistical mechanics implemented in the scientific software Gaussian16 [14] and we predict that all complexes can be formed with the exception of the FH and CO2 molecules, with N2 and ClH complexes with indeed very low, though negative, free energies of formation.
The tetrel C···X interactions in all complexes have been thoroughly studied by means of AIM and ELF methods hence defining the type of bond and interaction, ranging from very polarised bonds, with one electron in the C···X moiety, to intermadiate cases as in the carbenes :CH2 and :CF2 and silane :SiH2, with one and a half electrons in the C···X region.
The existence of known tetrel complexes of the carbonium ylide CB11H11 with amino derivatives, including pyridine, opens the door toward further experimental and theoretical studies in the electronic structure of unusual bonds and interactions between C(ylide) centers in carboranes and other atoms.
We hope that the results from this work can be used for the isolation of reactive species, such as the recently found dianion derived from proton extraction in the well known carborane anion CB11H12 () , a key molecule in the description of 3D aromaticity within boron chemistry.